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<p>PROBABILITY, STATISTICS,</p><p>AND DECISION FOR CIVIL</p><p>ENGINEERS</p><p>PROBABILITY, STATISTICS,</p><p>AND DECISION FOR CIVIL</p><p>ENGINEERS</p><p>Jack R. Benjamin</p><p>and</p><p>C. Allin Cornell</p><p>DOVER PUBLICATIONS, INC.</p><p>Mineola, New York</p><p>Copyright</p><p>Copyright © 1970 by Jack R. Benjamin and C. Allin Cornell</p><p>All rights reserved.</p><p>Bibliographical Note</p><p>This Dover edition, first published in 2014, is an unabridged republication</p><p>of the work originally published by McGraw-Hill, Inc., New York, in 1970.</p><p>Library of Congress Cataloging-in-Publication Data</p><p>Benjamin, Jack R. (Jack Ralph), 1917–1998, author.</p><p>Probability, statistics, and decision for civil engineers / Jack R.</p><p>Benjamin, C. Allin Cornell. — Dover edition.</p><p>pages cm</p><p>Reprint of: New York : McGraw-Hill, 1970.</p><p>Summary: “This text covers the development of decision theory and</p><p>related applications of probability. Extensive examples and illustrations</p><p>cultivate students’ appreciation for applications, including strength of</p><p>materials, soil mechanics, construction planning, and water-resource</p><p>design. Emphasis on fundamentals makes the material accessible to</p><p>students trained in classical statistics and provides a brief introduction to</p><p>probability. 1970 edition”— Provided by publisher.</p><p>eISBN-13: 978-0-486-31570-6</p><p>1. Probabilities. 2. Mathematical statistics. 3. Bayesian statistical</p><p>decision theory. I. Cornell, C. Allin, author. II. Title.</p><p>TA340.B45 2014</p><p>519.5—dc23</p><p>2014012854</p><p>Manufactured in the United States by Courier Corporation</p><p>78072401 2014</p><p>www.doverpublications.com</p><p>http://www.doverpublications.com/</p><p>To Sarah and Elizabeth</p><p>Preface</p><p>This book is designed for use by students, practitioners, teachers, and</p><p>researchers in civil engineering. Most books on probability and statistics</p><p>are intended for readers from many fields, and therefore the question might</p><p>be asked, “Why is it necessary to have a book on applied probability and</p><p>statistics for civil engineers alone?” Further, within civil engineering itself,</p><p>we might ask, “Why is it reasonable to attempt to write to an audience with</p><p>such widely varying needs?” The need for writing such a book is a result of</p><p>the unusual status of probability and statistics in civil engineering.</p><p>Although virtually all forward-looking civil engineers see the rationality</p><p>and utility of probabilistic models of phenomena of interest to the</p><p>profession, the number of civil engineers trained in probability theory has</p><p>been limited. As a result there has not yet been widespread adoption of</p><p>such models in practice or even in university courses below the advanced</p><p>graduate level. Indeed, there has not yet been sufficient development of</p><p>such models to permit a unified probabilistic approach to the many aspects</p><p>of the strength of materials, soil mechanics, construction planning, water-</p><p>resource design, and many other subjects where the methods could clearly</p><p>be useful.</p><p>Many departments of civil engineering now require that their students study</p><p>probability or statistics. Subsequent undergraduate technical courses,</p><p>however, usually lack material that draws upon that study, and this often</p><p>leaves the student without an appreciation for possible implications of the</p><p>theory. Through the use of illustrations and problems taken solely from the</p><p>civil-engineering field this book is designed to develop this appreciation for</p><p>applications. It can serve as a primary text for a course within a civil-</p><p>engineering department or as a supplementary text for courses taught in</p><p>other departments. The major compromise here was the acceptance of the</p><p>risk of obscuring or confusing the basic theory in describing the illustrative</p><p>physical problem. We could have avoided this risk by using only examples</p><p>dealing with coins, dice, red and black balls, etc., but the desire to develop</p><p>the ability to construct mathematical models of physical phenomena in</p><p>civil-engineering applications weighed heavily in favor of professional</p><p>illustrations.</p><p>Students finishing such a professionally oriented probability course may or</p><p>may not go on to advanced courses in the theory. This consideration</p><p>dictated at least an introductory coverage of a wide variety of material</p><p>coupled with a thorough treatment of the fundamentals. The kinds of</p><p>compromises involved in trying to decide between breadth in subject matter</p><p>and depth in fundamentals are obvious. It is unlikely that our many</p><p>decisions of this kind will be judged optimal by all, or, indeed, by ourselves</p><p>in time.</p><p>This emphasis on a thorough understanding of fundamentals, on a broad</p><p>coverage of subject matter, and on illustrations drawn from familiar</p><p>practice are appropriate for a book aimed at a growing number of practicing</p><p>civil engineers, researchers, and students who have an interest in teaching</p><p>themselves the elements of applied probability and statistics. Here the</p><p>continued exposure to professionally based illustrations, as simplified as</p><p>they may be, can help to sustain that interest and stimulate ideas for</p><p>developments of direct personal concern. (Although it is not a basic aim,</p><p>the usefulness of the text as a reference book is enhanced by a number of</p><p>convenient tables.)</p><p>A book designed for a broad audience of civil engineers is also important</p><p>because the training of civil engineers familiar with probability and</p><p>statistics is presently widely varied in depth, breadth, and source. Many</p><p>misuses and misinterpretations of probability and statistics exist in practice</p><p>and in the profession’s literature. The emphasis in this book on depth in</p><p>fundamentals is again critical. In a number of places in the book a</p><p>significant effort has been made to place additional emphasis on commonly</p><p>misunderstood points. To the beginning reader the effort and words may</p><p>seem excessive, but it is hoped that these discourses will serve as warnings</p><p>to the reader to be particularly alert when similar situations are</p><p>encountered. Readers with some experience in applications of probability</p><p>and statistics will recognize these subjects and perhaps benefit by the</p><p>additional discussion. It is hoped in particular that readers who have studied</p><p>only statistics in the past will appreciate the opportunity to study the</p><p>probability theory upon which statistics is based. A common reaction to</p><p>such a study is, on one hand, surprise at the degree of underlying unity in</p><p>statistical methods and, on the other, a new appreciation for the limitations</p><p>of classical statistics.</p><p>The most significant consideration in our decision to develop a book for</p><p>civil engineers of diverse backgrounds is the result of a major change</p><p>within the theory of applied probability and statistics. A new, still</p><p>somewhat controversial theory is being developed around a framework of</p><p>economic decision making. The implications to civil engineers are many. In</p><p>place of earlier emphasis on obtaining proper, objective descriptions of</p><p>repetitive physical phenomena, the new concern is with making decisions</p><p>involving economic gains and losses when uncertainty exists in the</p><p>decision maker’s mind regarding the state of nature. This new emphasis,</p><p>with its new interpretations and its new methods, is far more appropriate</p><p>and natural for civil engineers, whose profession is more closely involved</p><p>than any other in the economical design of one-of-a-kind systems subject to</p><p>the uncertain demands of natural and man-made environmental factors.</p><p>The new development (associated variously with such phrases as Bayesian</p><p>statistical decision theory, subjective probability, and utility theory) is, in</p><p>the authors’ minds, of such central importance that convenient access to it</p><p>must be developed for civil engineers. Even researchers and teachers well</p><p>versed in traditional probability and classical statistics can benefit from</p><p>exposure to decision theory. They will find a number of very reasonable</p><p>new ideas and a more liberal interpretations of some old ideas (including,</p><p>in a sense, justification for some natural “mistakes” they may very well</p><p>have been making in applying classical statistics).</p><p>of probability is</p><p>frequently called a random event. The following conditions must hold on the</p><p>probabilities assigned to the events in the sample space:</p><p>Axiom I The probability of an event is a number greater than or equal to</p><p>zero but less than or equal to unity:</p><p>images</p><p>Axiom II The probability of the certain event S is unity:</p><p>images</p><p>where S is the event associated with all the sample points in the sample</p><p>space.</p><p>Axiom III The probability of an event which is the union of two mutually</p><p>exclusive events is the sum of the probabilities of these two events:</p><p>images</p><p>Since S is the union of all simple events, the third axiom implies that Axiom</p><p>II could be written:</p><p>images</p><p>in which the Ei are simple events associated with individual sample points.‡</p><p>The first two axioms are simply convenient conventions. All probabilities</p><p>will be positive numbers and their sum over the simple events (or any</p><p>mutually exclusive, collectively exhaustive set of events) will be normalized</p><p>to 1. These are natural restrictions on probabilities which arise from</p><p>observed relative frequencies. If there are k possible outcomes to an</p><p>experiment and the experiment is performed M times, the observed relative</p><p>frequencies Fi are</p><p>images</p><p>where n1, n2, . . ., nk are the numbers of times each particular outcome was</p><p>observed. Since n1 + n2 + · · · + nk = M, each frequency satisfies Axiom I</p><p>and their sum satisfies Axiom II:</p><p>images</p><p>If, alternatively, one prefers to think of probabilities as weights on events</p><p>indicative of their relative likelihood, then Axioms I and II simply demand</p><p>that, after assigning to the set of all simple events a set of relative weights,</p><p>one normalizes these weights by dividing each by the total.</p><p>Axiom III is equally acceptable. It requires only that the probabilities are</p><p>assigned to events in such a way that the probability of any event made up of</p><p>two mutually exclusive events is equal to the sum of the probabilities of the</p><p>individual events. If the original assignment of probabilities is made on a set</p><p>of collectively exhaustive, mutually exclusive events, such as the set of all</p><p>simple events, there can be no possibility of violating this axiom. If, for</p><p>example, the source of these assignments is a set of observed relative</p><p>frequencies, as long as the original set of k outcomes has been properly</p><p>defined (in particular, not overlapping in any way), the relative frequency of</p><p>the outcome i or j is</p><p>images</p><p>Similarly, if an engineer assigns relative weights to a set of possible distinct</p><p>outcomes, he would surely be inconsistent if he felt that the relative</p><p>likelihood of either of a pair of disjoint outcomes was anything but the sum</p><p>of their individual likelihoods.</p><p>Suppose that the soils engineer in the above example decides that the odds</p><p>on there being clay soil at a depth of 30 ft are 1 to 3 (or 1 in 4), and sand is</p><p>just as likely; and if neither of these is present, the material will surely be</p><p>sound rock. The implication is that he gives clay a relative weight of 1 and</p><p>the other outcomes a total weight of 3. These possible outcomes include</p><p>only sand, with a weight 1, and rock, with weight 3 – 1, or 2. To be used as</p><p>probabilities these weights need normalizing by their sum, 4, to satisfy the</p><p>first and second axioms of probability. Let the event C be “there is clay 30 ft</p><p>below the footing” and let the events S and R be associated with the</p><p>presence of sand and rock, respectively. Then</p><p>images</p><p>Notice that the three axioms are satisfied and that</p><p>images</p><p>Once measures have been assigned in accord with these three axioms to the</p><p>points in the sample space, these probabilities may be operated on in the</p><p>manner to be demonstrated throughout this book. The engineer may be fully</p><p>confident that his results will be mathematically valid, but it cannot be</p><p>emphasized too strongly that the physical or practical significance of these</p><p>results is no better than the “data,” the assigned probabilities, upon which</p><p>they are based.</p><p>2.1.3 Simple Probabilities of Events</p><p>Certain relationships among the probabilities of events follow from the</p><p>relationships among events and from the axioms of probability. Many simple</p><p>conclusions are self-evident; others require some derivation. Further, some</p><p>additional relationships between events are defined in terms of relationships</p><p>between their probabilities. The remainder of this section treats and</p><p>illustrates these various relationships.</p><p>Probability of an event Since, in general, an event is associated with one or</p><p>more sample points or simple events, and since these simple events are</p><p>mutually exclusive by the construction of the sample space, the probability</p><p>of any event is the sum of the probabilities assigned to the sample points</p><p>with which it is associated. If an event contains all the sample points with</p><p>nonzero probabilities, its probability is 1 and it is sure to occur. If an event is</p><p>impossible, that is, if it cannot happen as the result of the experiment, then</p><p>the probabilities of all the sample points associated with the event are zero.</p><p>images</p><p>Fig. 2.1.10 Venn diagram. Illustrates decomposition of two events A and B</p><p>into mutually exclusive events A0,B0 and A ∩ B.</p><p>Probability of union The probability of an event which is the union of two</p><p>events A and B, disjoint or not, can be derived from the material in the</p><p>previous sections. The event A can be considered as the union of the</p><p>intersection A ∩ B and a nonoverlapping set of sample points, say, A0.</p><p>Similarly, event B is the union of two mutually exclusive events A ∩ B and</p><p>B0. These events are illustrated in Fig. 2.1.10.† By Axiom III</p><p>images</p><p>images</p><p>Now A imagesB can be divided into three mutually exclusive events, A0,</p><p>B0, and A ∩ B.</p><p>Therefore, by Axiom III,</p><p>images</p><p>Solving Eqs. (2.1.1a) and (2.1.1b) for P[A0] and P[B0] and substituting into</p><p>Eq. (2.1.2),</p><p>images</p><p>images</p><p>Fig. 2.1.11 Illustration of industrial-park utilities.</p><p>In words, the probability of the occurrence of either one event or another or</p><p>both is the sum of their individual probabilities minus the probability of their</p><p>joint occurrence. This extremely important result is easily verified</p><p>intuitively. In summing the probabilities of the events A and B to determine</p><p>the probability of a compound event A images B, one has added the</p><p>probability measure of the sample points in the event A ∩ B twice. In the</p><p>case of mutually exclusive events, when the intersection A ∩ B contains no</p><p>sample points, P[A ∩ B] = 0 and the equation reduces to Axiom III.</p><p>To illustrate this and following concepts, let us consider the design of an</p><p>underground utilities system for an industrial park containing six similar</p><p>building sites (Fig. 2.1.11). The sites have not yet been leased, and so the</p><p>nature of occupancy of each is not known. If the engineer provides water</p><p>and power capacities in excess of the demand actually encountered, he will</p><p>have wasted his client’s capital; if, on the other hand, the facilities prove</p><p>inadequate, expensive changes will be required. For simplicity, consider any</p><p>particular site and assume that the electric power required by the occupant</p><p>will be either 5 or 10 units, while the water capacity demanded will be either</p><p>1 or 2 units. Then the sample space describing an experiment associated</p><p>with a single occupant consists of four points, labeled (5,1), (10,1), (5,2), or</p><p>(10,2), according to the combination of levels of power and water</p><p>demanded. The space can be illustrated in either of two ways, as shown in</p><p>Fig. 2.1.12. The client, in an interview with the engineer, makes a series of</p><p>statements about odds and relative weights from which the engineer</p><p>calculates the following set of probabilities.</p><p>.</p><p>images</p><p>Fig. 2.1.12 Alternate graphical representations of the utilities-demand</p><p>sample space.</p><p>images</p><p>The probability of an event W2, “the water demand is 2 units” is the sum of</p><p>the probabilities of the corresponding, mutually exclusive, simple events.</p><p>images</p><p>Also, the probability of power demand being 10 units at a particular site is</p><p>images</p><p>The probability</p><p>that either the water demand is 2 units or the power demand</p><p>is 10 units may be calculated by Eq. (2.1.3).</p><p>images</p><p>or, since the intersection of events E10 and W2 is the simple event E10W2,</p><p>images</p><p>Notice that the same result is obtained by summing the probabilities of the</p><p>simple events in which one observes either a water demand of 2 units or a</p><p>power demand of 10 units or both.</p><p>images</p><p>Conditional probability A concept of great practical importance is</p><p>introduced into the axiomatic theory of probability through the following</p><p>definition. The conditional probability of the event A given that the event B</p><p>has occurred, denoted P[A | B], is defined as the ratio of the probability of</p><p>the intersection of A and B to the probability of the event B.</p><p>images</p><p>(If P[B] is zero, the conditional probability P[A | B] is undefined.)</p><p>The conditional probability can be interpreted as the probability that A has</p><p>occurred given the knowledge that B has occurred. The condition that B has</p><p>occurred restricts the outcome to the set of sample points in B, or the</p><p>conditional sample space, but should not change the relative likelihoods of</p><p>the simple events in B. If the probability measure of those points in B that</p><p>are also in A, P[A ∩ B], is renormalized by the factor 1/P[B] to account for</p><p>operation within this reduced sample space, the result is the ratio P[A ∩</p><p>B]/P[B] for the probability of A given B.</p><p>In the preceding illustration the engineer might have need for the probability</p><p>that a site with a power demand of E10 will also require a water demand W2.</p><p>In this case</p><p>images</p><p>In applications, P[B] and P[A | B] often come from a study of the problem,</p><p>whereas actually the joint probability P[A ∩ B] is desired; this is obtained as</p><p>follows:</p><p>images</p><p>Where many events are involved, the following expansion is often helpful:</p><p>images</p><p>For example, for three events</p><p>images</p><p>Illustrations are given later in the section and in problems.</p><p>Independence If two physical events are not related in any way, we would</p><p>not alter our measure of the probability of one even if we knew that the other</p><p>had occurred. This intuitive notion leads to the definition of probabilistic (or</p><p>stochastic) independence. Two events A and B are said to be independent if</p><p>and only if</p><p>images</p><p>From this definition and Eq. (2.1.4a), the independence of events A and B</p><p>implies that</p><p>images</p><p>images</p><p>images</p><p>Any of these equations can, in fact, be used as a definition of independence.</p><p>Within the mathematical theory, one can only prove independence of events</p><p>by obtaining P[A], P[B], and P[A ∩ B] and demonstrating that one of these</p><p>equations holds. In engineering practice, on the other hand, one normally</p><p>relies on knowledge of the physical situation to declare that in his model two</p><p>particular events shall (or shall not) be assumed independent. From the</p><p>assumption of independence, the engineer can calculate one of the three</p><p>quantities, say, P[A ∩ B], given the other two.</p><p>In general, events A, B, C, . . ., N are mutually independent if and only if</p><p>images</p><p>This is the theorem known as the multiplication rule. In words, if events are</p><p>independent, the probability of their joint occurrence is simply the product</p><p>of their individual probabilities of occurrence.</p><p>Returning to the industrial-park illustration, let us assume that the engineer,</p><p>not content with the loose manner in which his client assigned probabilities</p><p>to demands, has sampled a number of firms in similar industrial parks. He</p><p>has concluded that there is no apparent relationship between their power and</p><p>water demands. A high power demand, for example, does not seem to be</p><p>correlated with a high water demand.</p><p>Based on information easily obtained from the respective utility companies,</p><p>the engineer assigns the following probabilities:</p><p>images</p><p>Adopting the assumption of stochastic independence of the water and power</p><p>demands, † the engineer can calculate the following probabilities for the</p><p>joint occurrences or simple events.</p><p>images</p><p>Decisions under uncertainty With no more than the elementary operations</p><p>introduced up to this point, we can demonstrate the usefulness of</p><p>probabilistic analysis when the engineer must make economic decisions in</p><p>the face of uncertainty. We shall present only the rudiments of decision</p><p>analysis at this point. We chose to continue to use the preceding example to</p><p>introduce these ideas. For simplicity of numbers, let us concentrate on the</p><p>water demand only and investigate the design capacity of a secondary main</p><p>serving a pair of similar sites in the industrial park. The occupancies of the</p><p>two sites represent two repeated trials of the experiment described above.</p><p>Denote the event that the demand of each firm is one unit by W1W1 and the</p><p>event that the demand of the first is one unit and the second is two units by</p><p>W1W2, and so forth. Assuming stochastic independence of the demands</p><p>from the two sites, one can easily calculate the probabilities of various</p><p>combinations of outcomes.</p><p>images</p><p>Notice that events W1W2 and W2W1 lead to the same total demand; on the</p><p>sample space of this two-site experiment, we could define new events D2,</p><p>D3, and D4 which correspond to total demands of two, three, and four units,</p><p>respectively (see Fig. 2.1.13). It is most often the case that engineers are</p><p>interested in outcomes which have associated numerical values. This</p><p>observation is expanded upon in Sec. 2.2.</p><p>The assumption of independence between sites implies that the engineer</p><p>feels that there would be no reason to alter the probabilities of the demand of</p><p>the second firm if he knew the demand of the first. That is, knowledge of the</p><p>demand of the first gives the engineer no new information about the demand</p><p>of the second. Such might be the case, for example, if the management of</p><p>the second firm chose its site without regard for the nature of its neighbor. If</p><p>the demands of all six sites are mutually independent, the probability that all</p><p>the sites will demand two units of water is:</p><p>images</p><p>How can such information be put to use to determine the best choice for the</p><p>capacity of the secondary pipeline? These estimates of the relative</p><p>likelihoods must in some way be related to the relative costs of the designs</p><p>and the losses associated with possible inadequacies. Suppose the engineer</p><p>has compiled the following cost table:</p><p>images</p><p>Fig. 2.1.13 Two-site water-demand illustration sample space.</p><p>Initial costs:</p><p>images</p><p>Cost associated with possible later enlargement:</p><p>images</p><p>A common method used to account for uncertain losses is to weight the</p><p>possible losses by the probabilities of their occurrences. Thus to the initial</p><p>cost of a moderate design of capacity three units the engineer might add the</p><p>weighted loss (0.49) ($1500) = $735, which is associated with a possible</p><p>need for later enlargement if both firms should demand two units (the event</p><p>W2W2). If a two-unit capacity is chosen, either of two future events, D3 or</p><p>D4, will lead to additional cost. The weighted cost of each contributes to the</p><p>total expected cost of this design alternative. These costs are called</p><p>“expected costs” for reasons which will be discussed in Sec. 2.4. The</p><p>validity of their use in making decisions, to be discussed more fully in Chap.</p><p>5,† will be accepted here as at least intuitively satisfactory. The following</p><p>table of expected costs (over the basic initial cost of the two-unit capacity)</p><p>can be computed for each alternative design.</p><p>images</p><p>A design providing an initial capacity of three units appears to provide the</p><p>best compromise between initial cost and possible later expenses. Notice</p><p>that the common cost of two units does not enter the decision of choosing</p><p>among the available alternatives.</p><p>As modifications lead to future rather than initial expense, the effect of the</p><p>introduction of interest rates, and hence the time value of money, into the</p><p>economic analysis would be to increase the relative merit of this strategy</p><p>(three units) with respect to the large-capacity (four-unit) design while</p><p>decreasing its relative advantage over</p><p>the small-capacity (two-unit) system.</p><p>A high interest rate could make the latter design more economical.</p><p>To design the primary water lines feeding all sites in the industrial park, a</p><p>similar but more complex analysis is required. In dealing with repeated trials</p><p>of such two-outcome experiments, one is led to the binomial model to be</p><p>introduced in Sec. 3.1.2. Where more than two demand levels or more than</p><p>one service type are needed, the multinomial distribution (Sec. 3.6.1) will be</p><p>found to apply.</p><p>In Chaps. 5 and 6 we shall discuss decision making in more detail. The</p><p>following two illustrations demonstrate further the computations of</p><p>probabilities of events using the relationships we have discussed up to this</p><p>point, and the use of expected costs in decisions under uncertainty.</p><p>Illustration: Construction scheduling A contractor must select a strategy</p><p>for a construction job. Two independent operations, I and II, must be</p><p>performed in succession. Each operation may require 4, 5, or 6 days to</p><p>complete. A sample space is illustrated in Fig. 2.1.14. M4 is the event that</p><p>operation I requires 4 days, N4 that II requires 4 days, etc. Each operation</p><p>can be performed at three different rates, each at a different cost, and each</p><p>leading to different time requirement likelihoods. In addition, if the job is</p><p>not completed in 10 days, the contractor must pay a penalty of $2000 per</p><p>day. The total time required for each combination of time requirements is</p><p>shown in parentheses in Fig. 2.1.14. For example, M4 ∩ N6 requires a total</p><p>of 10 days.</p><p>images</p><p>Fig. 2.1.14 Sample space for construction-strategy illustration.</p><p>Table 2.1.1</p><p>images</p><p>The contractor judges from experience that by working at rate A his</p><p>probability of completing phase I in 4 days, event M4, is only 0.2, in 5 days</p><p>(or M5) is 0.5, and in 6 days (M6) is 0.3. Proceeding in this manner, he</p><p>assigns a complete set of probabilities (Table 2.1.1) to all possibilities,</p><p>reflecting that he can probably accelerate the job by working at a more</p><p>costly rate. He assumes that the events M4, M5, and M6 are independent of</p><p>N4, N5, and N6.</p><p>The expected costs E[cost] of construction can now be calculated.</p><p>For level I at rate A,</p><p>images</p><p>or</p><p>images</p><p>Similarly,</p><p>images</p><p>Similarly, all expected costs of construction are:</p><p>images</p><p>The optimum strategy is rate A for operation I and rate D for II to</p><p>obtain a minimum expected cost of construction.</p><p>The possibility of an overtime penalty must also be included in the</p><p>total cost. The probability of strategy AD requiring 8 days of time is the</p><p>probability of (M4 ∩ N4) and, owing to independence, is simply the product</p><p>of the individual probabilities of each operation requiring 4 days. Assuming</p><p>independence of the events, under strategy AD,</p><p>images</p><p>A 9-day construction time can occur in two mutually exclusive ways:</p><p>4 days required for I and 5 days for II, or 5 days for I and 4 days for II. This</p><p>event is crosshatched in Fig. 2.1.14.</p><p>images</p><p>Under strategy AD, P[9 days] = (0.2)(0.4) + (0.5)(0.1) = 0.13.</p><p>Similarly, a 10-day time occurs as shown in Fig. 2.1.14 and, with the</p><p>probabilities of strategy AD, has probability</p><p>images</p><p>Losses are associated with construction times of 11 or 12 days. Using</p><p>rates A and D,</p><p>images</p><p>The expected penalty, if strategy AD is adopted, is, then,</p><p>images</p><p>The complete penalty results are shown in Table 2.1.2, and the total</p><p>expected costs, that is, those due to construction plus those due to possible</p><p>penalties, are collected in Table 2.1.3.</p><p>Table 2.1.2</p><p>images</p><p>Table 2.1.3</p><p>images</p><p>The optimum strategy has a minimum expected total cost and is in fact</p><p>strategy CD.</p><p>Expected costs simply provide a technique for choosing among</p><p>alternative strategies. The contractor’s cost will not be $2712 if he chooses</p><p>strategy CD, but rather the cost of either 4, 5, or 6 days with I and with II</p><p>plus $2000 per actual day of overtime.</p><p>Illustration: Analysis of bridge lifetimes† As an example of the</p><p>construction of a more complex probability model from simple basic</p><p>assumptions, consider this problem in bridge design. Assume that a bridge</p><p>or culvert is usually replaced either because a flood exceeding the capacity</p><p>of the structure has occurred or because it becomes obsolete owing to a</p><p>widening or rerouting of the highway. The designer is interested in the</p><p>likelihood that the life of the structure will come to an end in each of the</p><p>years after construction.‡ Assume that there is a constant probability p that</p><p>in any year a flow exceeding the capacity of the culvert will take place. Let</p><p>ri be the probability that the structure will become obsolete in year i given</p><p>that it has not become obsolete prior to year i. For most situations this</p><p>probability grows with time. It may well be a reasonable engineering</p><p>assumption that the effects of floods and obsolescence are unrelated and that</p><p>the occurrences of critical flood magnitudes from year to year are</p><p>independent events. Our problem is to determine the probability that the life</p><p>of the structure comes to an end in year j for the first time.</p><p>Each year is a simple experiment with events defined as</p><p>images</p><p>The elementary events in these simple experiments are</p><p>images</p><p>The probability that the structure’s life does not end in the first year</p><p>is, owing to the assumed independence, †</p><p>images</p><p>Successive years represent repetitions of such experiments. The</p><p>probability that the life does not end in either of the first two years is, by Eq.</p><p>(2.1.4a),</p><p>images</p><p>Similarly, the last term on the right-hand side can be written</p><p>images</p><p>This is easily verified by writing out the definitions of the conditional</p><p>probabilities involved, although a simple reading of the statement suggests</p><p>why it must be so.</p><p>Because of the various assumptions of independence made above,</p><p>images</p><p>Putting these results together,</p><p>images</p><p>Clearly, the probability that the structure survives floods and</p><p>obsolescence through j years is</p><p>images</p><p>which by simple extension of the argument above is</p><p>images</p><p>For the structure’s life to first come to end in year j, on the other</p><p>hand, it must have survived j – 1 years, which will have happened with</p><p>probability</p><p>images</p><p>and must then either have become obsolete or met a critical flood in</p><p>year j. The latter event, Aj images Bj, has probability, given previous</p><p>survival, of</p><p>images</p><p>Equation (2.1.3) applies, subject to the conditioning event, previous</p><p>survival, and</p><p>images</p><p>Owing to the various independences,</p><p>images</p><p>Finally, then,</p><p>images</p><p>For example, if the structure is designed for the so-called “50-year</p><p>flood,” † it implies that p = 1/50 = 0.02, and, if ri = 1 – e–0.025i, i = 1, 2, 3, . .</p><p>., then</p><p>images</p><p>A plot of these probabilities for the years j = 1 to 22 is given in Fig.</p><p>2.1.15.</p><p>Combined with economic data these probabilities would permit the</p><p>engineer to calculate an expected present worth of this design, to be</p><p>compared with those of other alternate designs of different capacities for</p><p>flow and perhaps with different provisions to reduce the likelihood of</p><p>obsolescence.</p><p>Total probability theorem The equation defining conditional probabilities,</p><p>Eq. (2.1.4a), can be manipulated to yield another important result in the</p><p>probability of events. Given a set of mutually exclusive, collectively</p><p>exhaustive events, B1, B2, . . ., Bn, one can always expand the probability</p><p>P[A] of another event A in the following manner:</p><p>images</p><p>images</p><p>Fig. 2.1.15 Solution to culvert-life example. Probabilities of finding various</p><p>lengths of life are plotted against length of life.</p><p>Figure 2.1.16 illustrates this fact. Each term in the sum can be expanded</p><p>using Eq. (2.1.4a):</p><p>images</p><p>This result is called “the theorem of total probabilities.” It represents the</p><p>expansion of</p><p>the probability of an event in terms of its conditional</p><p>probabilities, conditioned on a set of mutually exclusive, collectively</p><p>exhaustive events. It is often a useful expansion to consider in problems</p><p>when it is desired to compute the probability of an event A, since the terms</p><p>in the sum may be more readily obtainable than the probability A itself.</p><p>images</p><p>Fig. 2.1.16 Venn diagram for total probability theorem. Event A intersects</p><p>mutually exclusive and collectively exhaustive events Bi.</p><p>Illustration: Additive random demands on engineering systems Consider</p><p>the generalized civil-engineering design problem of providing “capacity” for</p><p>a probabilistic “demand.” Depending on the situation, demand may be a</p><p>loading, a flood, a peak number of users, etc., while the corresponding</p><p>capacity may be that of a building, a dam, or a highway. In many examples</p><p>to follow, the general terminology, that is, demand and capacity, will be used</p><p>with the express purpose of encouraging the reader to supply his preferred</p><p>specific application. In this example two possible types (primary and</p><p>secondary) of capacity at different unit costs are available, and loss is</p><p>incurred if demand exceeds the total capacity provided or if it requires the</p><p>use of some secondary capacity. This situation faces every designer, for the</p><p>peak demand is often uncertain and the design usually cannot economically</p><p>be made adequate for the maximum possible demand. The engineer seeks a</p><p>balancing of initial cost and potential future losses.</p><p>For example, a building frame should be able to sustain a moderate</p><p>seismic load without visible damage to the structure. During a rare, major</p><p>earthquake, however, a properly designed structure will develop secondary</p><p>resistance involving large plastic deformations and some acceptable level of</p><p>damage to windows and partitions. Design for zero damage under all</p><p>possible conditions is impossible or uneconomical. Similar problems arise in</p><p>design of systems in which provision for future expansion is included. The</p><p>future demand is unknown, so that the estimation of the optimum funds to</p><p>be spent now to provide for expansion in the future is a similar type of</p><p>capacity-demand situation.</p><p>Assume, in this example, that demand arises from two additive</p><p>sources A and B and that the engineer assigns probabilities as shown in</p><p>Table 2.1.4 to the various levels from source B and conditional probabilities</p><p>to various levels from source A for each level of source B. The two sources</p><p>are not independent.</p><p>Table 2.1.4 Conditional probabilities of [A, i Bj] and the probabilities of</p><p>Bj</p><p>images</p><p>The probabilities of the A levels can be found by applying Eq. (2.1.12)</p><p>for each level from source A. For example,</p><p>images</p><p>Similarly,</p><p>images</p><p>Before proceeding with the analysis, note that the engineer has</p><p>assigned some zero probabilities.† Two interpretations exist of a zero</p><p>probability. First, the event may simply be impossible. Second, it may be</p><p>possible, but its likelihood negligible, and the engineer is willing to</p><p>effectively exclude this level from the current study (while retaining the</p><p>option to assign later a nonzero probability).</p><p>A two-dimensional sample space for the demand levels appears in</p><p>Fig. 2.1.17. Any simple event is an intersection Ai ∩ Bj of an A level and B</p><p>level. The various simple events which lead to the same total demand Dk are</p><p>also indicated in this figure, the total values being given in parentheses. To</p><p>determine the probabilities of an event such as D700 = [total demand is 700],</p><p>we find the probability of this event as the union of mutually exclusive,</p><p>simple events (and hence as simply the sum of their probabilities). Example:</p><p>images</p><p>images</p><p>Fig. 2.1.17 Demand-level sample space showing how various total demands</p><p>may arise.</p><p>A similar calculation for D500 involves four terms:</p><p>images</p><p>Similarly, we find that the probabilities of various demand levels are:</p><p>images</p><p>Although questions about the events which lead to particular demand</p><p>levels are not asked in this decision problem, they might well be of interest</p><p>in such problems. We ask one such question here to illustrate conditional</p><p>sample spaces and conditional probabilities. What is the conditional sample</p><p>space given that the total demand is 600? It is the set of those events which</p><p>produce such a total demand, namely A200 ∩ B400, A300 ∩ B300, and A400 ∩</p><p>B200. This space is illustrated in Fig. 2.1.18. The conditional probabilities of</p><p>these events, given total demand is 600, are</p><p>images</p><p>images</p><p>Fig. 2.1.18 Conditional sample space for a given total demand of 600.</p><p>Table 2.1.5 Example: expected cost calculations for several possible</p><p>designs</p><p>images</p><p>Because D600 includes the event A400 ∩ B200, we get</p><p>images</p><p>Notice that the unconditional probabilities of these events are 0.04,</p><p>0.02, and 0. These conditional probabilities are just these same relative</p><p>values normalized to sum to unity.</p><p>Assume that primary capacity can be provided at $1000 per unit and</p><p>that the secondary capacity cost is $100 per unit. If the demand is such that</p><p>the secondary capacity must be used, the associated (“damage”) loss is</p><p>$1000 per unit used, and if the demand exceeds total (primary plus</p><p>secondary) capacity, the (“failure”) loss is $2000 per unit of excess over</p><p>total capacity. The design alternatives include any combination of 0 to 700</p><p>primary capacity and 0 to 700 secondary capacity.</p><p>Associated with each design alternative is a primary capacity cost</p><p>Cp, a secondary capacity cost Cs, and an expected cost associated with</p><p>potential loss due to excessive demands. The latter cost includes a</p><p>component due to demand possibly exceeding primary capacity, but not</p><p>secondary capacity, and a component arising due to demand possibly</p><p>exceeding the total capacity. {In the latter event, “failure,” no “damage” loss</p><p>is involved; e.g., if primary capacity is 500, secondary capacity is 100, and a</p><p>demand of 700 occurs, the loss is 2000[700 – (500 + 100)], not this plus</p><p>1000(700 – 500).}</p><p>A number of expected cost computations are illustrated in Table</p><p>2.1.5. The trend, as capacities are decreased from the most (initially) costly</p><p>design of 700 units of primary capacity, is toward lowering total cost by</p><p>accepting higher risks. After a point, the risks become too great relative to</p><p>the consequences and to the initial costs, and total expected costs rise again.</p><p>Bayes’ theorem Continuing the study of the event A and the set of events Bi</p><p>considered in Eq. 2.1.11 (Fig. 2.1.16), examine the conditional probability of</p><p>Bj given the event A. By Eq. (2.1.4a), and since, clearly,</p><p>images</p><p>The numerator represents one term in Eq. (2.1.11) and can be replaced as in</p><p>Eq. (2.1.12) by the product P[A | Bj]P[Bj], and the denominator can be</p><p>represented by the sum of such terms, Eq. (2.1.12). Substituting,</p><p>images</p><p>This result is known as Bayes’ theorem or Bayes’ rule. Its simple derivation</p><p>belies its fundamental importance in engineering applications. As will best</p><p>be seen in the illustration to follow, it provides the method for incorporating</p><p>new information with previous or, so-called prior, probability assessments</p><p>to yield new values for the engineer’s relative likelihoods of events of</p><p>interest. These new (conditional) probabilities are called posterior</p><p>probabilities. Bayes’ theorem will be more fully explained and applied in</p><p>Chaps. 5 and 6.</p><p>Illustration: Imperfect testing Bayes’ theorem can be generalized in</p><p>application by calling the unknown classification the state, and by</p><p>considering that some generalized sample has been observed. Symbolically,</p><p>Eq. (2.1.13) becomes</p><p>images</p><p>To illustrate the generalization, assume that an existing reinforced</p><p>concrete building is being surveyed to determine its adequacy for a new</p><p>future use. The engineer has studied the appearance and past performance of</p><p>the concrete and, based on professional judgment, decides that the concrete</p><p>quality can be classified as either 2000, 3000, or 4000 psi (based on the</p><p>usual 28-day cylinder strength). He also assigns relative likelihoods or</p><p>probabilities to these states:</p><p>images</p><p>Concrete cores are to be cut and tested to help ascertain the true</p><p>state. The engineer believes that a core gives a reasonably reliable</p><p>prediction, but that it is not conclusive. He consequently assigns numbers</p><p>reflecting the reliability of the technique in the form of conditional</p><p>probability measure on the possible core-strength values z1, z2, or z3 (in this</p><p>case, core strength† of, say, 2500, 3500, or 4500 psi) as predictors of the</p><p>unknown state:</p><p>P [core strength | state]</p><p>images</p><p>In words, if the true 28-day concrete quality classification is 3000</p><p>psi, the technique of taking a core will indicate this only 60 percent of the</p><p>time. The total error probability is 40 percent, divided between z1 and z3.</p><p>That is, the technique will significantly overestimate or underestimate the</p><p>true quality 4 times in 10, on the average. Controlled experiments using the</p><p>technique on concrete of known strength are used to produce such reliability</p><p>information.</p><p>A core is taken and found to have strength 2500 psi favoring a 28-</p><p>day strength of 2000 psi; that is, z1 is observed. The conditional probabilities</p><p>of the true strength are then [Eq. (2.1.13)]</p><p>images</p><p>The sample outcome causes the exclusion of 4000 as a possible state</p><p>and shifts the relative weights more towards the indicated state.</p><p>In the light of the test’s limitations, the engineer chooses to take a</p><p>sample of two independent cores. In this case, it makes no difference if the</p><p>calculation of posterior probabilities is made for each core in succession or</p><p>for both cores simultaneously. Consider the latter approach first. Assume</p><p>that the first core indicated z1 and the second core indicated z2. The</p><p>probability of finding the sample outcome {z1,z2} if the state is really 2000</p><p>(or 3000 or 4000) psi is the product of two conditional probabilities (since</p><p>the core results are assumed independent). Thus,</p><p>images</p><p>Recall that the probabilities of state prior to this sample of two cores</p><p>were 0.3, 0.6, and 0.1. The posterior probabilities then become</p><p>images</p><p>The role of Bayes’ theorem as an “information processor” is revealed</p><p>when it is recognized that the engineer might have taken the first core only,</p><p>found z1 (favoring 2000 psi), computed the posterior probabilities of state as</p><p>(0.635, 0.365, 0), and only then decided that another core was desirable. At</p><p>this point his prior probabilities (prior, now, only to the second core) are</p><p>0.635, 0.365, 0. The posterior probabilities, given that the second core</p><p>favored 3000, become</p><p>images</p><p>As they must, these probabilities are the same as those computed by</p><p>considering the two cores as a single sample. If a third core (or several more</p><p>cores) were taken next, these probabilities 0.47, 0.53, 0 would become the</p><p>new prior probabilities. Bayes’ theorem will permit the continuous up-dating</p><p>of the probabilities of state as new information becomes available. The</p><p>information might next be of some other kind, for example, the uncovering</p><p>of the lab data obtained from test cylinders cast at the time of construction</p><p>and tested at 28 days. Such information, of course, would have a different</p><p>set of conditional probabilities of sample given state (quite possibly, in this</p><p>case, with somewhat smaller probabilities of “errors,” that is, with smaller</p><p>probabilities of producing samples favoring a state other than the true one).</p><p>This simple example illustrates how all engineering sampling and</p><p>experimentation can better be viewed in a probabilistic formulation. For</p><p>reasons of expediency and economy, most testing situations measure a</p><p>quantity which is only indirectly related to the quantity that is of</p><p>fundamental engineering interest. In this example, the engineer has</p><p>measured an extracted core’s ultimate compressive strength in order to</p><p>estimate the concrete’s 28-day strength, which in turn is known to be</p><p>correlated with its compressive strength in bending strength, shear (or</p><p>diagonal tension) strength, corrosion resistance, durability, etc. The soils</p><p>engineer may measure the density of a compacted fill not because he wishes</p><p>to estimate its density, but because he wishes to estimate its strength and</p><p>ultimately the embankment’s stability. Various hardness-testing apparatuses</p><p>exist because testing for hardness is a simple, nondestructive way to</p><p>estimate the factor of direct interest, that is, a material’s strength.</p><p>Both in the actual making of the measurement and in the assumed</p><p>relationship between the measurable quantity (sample) and factor of direct</p><p>interest (state) there may be experimental error and variation, and hence</p><p>uncertainty. For example, accompanying a hardness-testing procedure is</p><p>usually a graph, giving a strength corresponding to an observed value of a</p><p>hardness measurement. But repeated measurements on a specimen may not</p><p>give the same hardness value, and, when the graph was prepared for a given</p><p>hardness there may have been some variation in the strength values about</p><p>the average value through which the graph was drawn. It is this uncertainty</p><p>which is reflected in the conditional probabilities P[sample | state]. A very</p><p>special case is that where there is negligible measurement error and where</p><p>the relationship between the sampled quantity and the factor of interest is</p><p>exact (for example, if the state can be measured directly). In this case, the</p><p>engineer will logically assign P[sample | state] = 1 if the sample value</p><p>indicates (i.e., “favors”) the state, and 0 if it does not. These assignments are</p><p>implicit in any nonprobabilistic model of testing. With these special</p><p>conditional probability assignments, inspection of Bayes’ theorem will</p><p>reveal that the only nonzero term in the denominator is the prior probability</p><p>of the state favored by the observed sample, while the numerator is the same</p><p>for this state and zero for all states not favored by the sample. Hence, if the</p><p>entire sampling procedure is, indeed, perfect, the posterior probabilities of</p><p>the states are zero except for the state favored or indicated by the sample.</p><p>This state has probability 1. If, however, there is any uncertainty in the</p><p>procedure, whether due to experimental error or inexact relationships</p><p>between predicted quantity and predicting quantity, at least some of the</p><p>other states will retain nonzero probabilities.† The better the procedure, the</p><p>more sharply centered will be its conditional probabilities on the sample-</p><p>favored state and the higher will be the, posterior probability of this state.</p><p>This fundamental engineering problem of predicting one quantity given an</p><p>observation of another related or correlated one will occur throughout this</p><p>text. In Secs. 2.2.2 and 2.4.3, and in Sec. 3.6.2, the probabilistic aspects of</p><p>the problem are discussed. In Sec. 4.3 the problem is treated statistically;</p><p>that is, questions of determining from observed data the “best” way to</p><p>predict one quantity given another are discussed. In Chaps. 5 and 6, we shall</p><p>return to the decision aspects of the problem.</p><p>2.1.4 Summary</p><p>In Sec. 2.1 we have presented the basic ideas of the probability of events.</p><p>After defining events and the among-events relationships of union,</p><p>intersection, collectively exhaustive, and mutually exclusive, we discussed</p><p>the assignment of probabilities to events. We found the following:</p><p>1. The probability of the union of mutually exclusive events was the sum of</p><p>their probabilities.</p><p>2. The probability of the union of two nonmutually exclusive events was the</p><p>sum of their probabilities minus the probability of their intersection.</p><p>3. The probability of the intersection of two events is the product of their</p><p>probabilities only if the two events are stochastically independent. In</p><p>general, a conditional probability must appear in the product.</p><p>These basic definitions were manipulated to obtain</p><p>a formula for the</p><p>probability of the intersection of several events, the total probability</p><p>theorem, and Bayes’ theorem.</p><p>The use of simple probability notions in engineering decisions was</p><p>illustrated, using the weighted or expected-cost criterion.</p><p>2.2 RANDOM VARIABLES AND DISTRIBUTIONS</p><p>Most civil-engineering problems deal with quantitative measures. Thus in</p><p>the familiar deterministic formulations of engineering problems, the</p><p>concepts of mathematical variables and functions of variables have proved</p><p>to be useful substitutes for less precise qualitative characterizations. Such is</p><p>also the case in probabilistic models, where the variable is referred to as a</p><p>random variable. It is a numerical variable whose specific value cannot be</p><p>predicted with certainty before an experiment. In this section we will first</p><p>discuss its description for a single variable, and then for two or more</p><p>variables jointly.</p><p>2.2.1 Random Variables</p><p>The value assumed by a random variable associated with an experiment</p><p>depends on the outcome of the experiment. There is a numerical value of the</p><p>random variable associated with every simple event defined on the sample</p><p>space, but different simple events may have the same associated value of the</p><p>random variable.† Every compound event corresponds to one or more or a</p><p>range of values of the random variable.</p><p>In most engineering problems there is seldom any question about how to</p><p>define the random variable; there is usually some “most natural” way. The</p><p>traffic engineer in the car-counting illustration (Sec. 2.1.1) would say, “Let X</p><p>equal the number of cars observed.” In other situations the random variable</p><p>in question might be Y, the daily discharge of a channel, or Z, stress at yield</p><p>of a steel tensile specimen. In fact, a random variable is usually the easiest</p><p>way to describe most engineering experiments. The cumbersome</p><p>subscripting of events found in previous illustrations could have been</p><p>avoided by dealing directly with random variables such as demand level,</p><p>concrete strength, etc., rather than with the events themselves.</p><p>The behavior of a random variable is described by its probability law, which</p><p>in turn may be characterized in a number of ways. The most common way is</p><p>through the probability distribution of the random variable. In the simplest</p><p>case this may be no more than a list of the values the variable can take on</p><p>(i.e., the possible outcomes of an experiment) and their respective</p><p>probabilities.</p><p>Discrete probability mass function (PMF) When the number of values a</p><p>random variable can take on is restricted to a countable number, the values</p><p>1, 2, 3, and 4, say, or perhaps all the positive integers, 0, 1, 2, . . ., the</p><p>random variable is called discrete, and its probability law is usually</p><p>presented in the form of a probability mass function, or PMF. This function</p><p>pX(x) of the random variable X‡ is simply the mathematical form of the list</p><p>mentioned above:</p><p>images</p><p>For example, having defined the random variable X to be the number of</p><p>vehicles observed stopped at the traffic light, the engineer may have</p><p>assigned probabilities to the events (Fig. 2.1.1) and corresponding values of</p><p>X such that</p><p>images</p><p>The probability mass function is usually plotted as shown in Fig. 2.2.1a,</p><p>with each bar or spike being proportional in height to the probability that the</p><p>random variable takes on that value.</p><p>To satisfy the three axioms of probability theory the probability mass</p><p>function clearly must fulfill three conditions:</p><p>images</p><p>images</p><p>images</p><p>The sums in Eqs. (2.2.2b) and (2.2.2c) are, of course, only over those values</p><p>of x where the probability mass function is defined.</p><p>Cumulative distribution function (CDF) An equivalent means† by which</p><p>to describe the probability distribution of a random variable is through the</p><p>use of a cumulative distribution function, or CDF. The value of this function</p><p>FX(x) is simply the probability of the event that the random variable takes on</p><p>value equal to or less than the argument:</p><p>images</p><p>For discrete random variables, i.e., those possessing probability mass</p><p>functions, this function is simply the sum of the values of the probability</p><p>mass function over those values less than or equal to x that the random</p><p>variable X can take on.</p><p>images</p><p>images</p><p>Fig. 2.2.1 Probability law for traffic illustration. (a) Probability mass</p><p>function PMF; (b) cumulative distribution function CDF.</p><p>The CDF of the random variable X, the number of stopped cars, with the</p><p>PMF described on page 73, is a step function:</p><p>images</p><p>Although clumsy to specify analytically, such functions are easy to</p><p>visualize. This discontinuous function is graphed in Fig. 2.2.1b. One would</p><p>read from it, for example, that the probability of finding a line containing</p><p>two or fewer vehicles is FX(2) = 0.6 [which equals pX(0) + pX(1) + pX(2) =</p><p>0.1 + 0.2 + 0.3] or that FX(3) = 0.8 [or FX(2) + pX(3) = 0.6 + 0.2].</p><p>The PMF can always be recovered if the CDF is given, since the former</p><p>simply describes the magnitudes of the individual steps in the CDF.</p><p>Formally,</p><p>images</p><p>where images is a small positive number.</p><p>Continuous random variable and the PDF Although the discrete random</p><p>variable is appropriate in many situations (particularly where items such as</p><p>vehicles are being counted), the continuous random variable is more</p><p>frequently adopted as the mathematical model for physical phenomena of</p><p>interest to civil engineering. Unlike the discrete variable, the continuous</p><p>random variable is free to take on any value on the real axis.† Strictly</p><p>speaking, one must be extremely careful in extending the ideas of sample</p><p>spaces to the continuous case, but conceptually the engineer should find the</p><p>continuous random variable more natural than the discrete. All the</p><p>engineer’s physical variables—length, mass, and time—are usually dealt</p><p>with as continuous quantities. A flow rate might be 1000, 1001, 1001.1, or</p><p>1001.12 cfs. Only the inadequacies of particular measuring devices can lead</p><p>to the rounding off that causes the measured values of such quantities to be</p><p>limited to a set of discrete values.</p><p>The problem of specifying the probability distribution (and hence the</p><p>probability law) of a continuous random variable X is easily managed. If the</p><p>x axis is separated into a large enough number of short intervals each of</p><p>infinitesimal length dx, it seems plausible that we can define a function fX(x)</p><p>such that the probability that X is in interval x to x + dx is fx(x) dx. Such a</p><p>function is called the probability density function, or PDF, of a continuous</p><p>random variable.</p><p>Since occurrences in different intervals are mutually exclusive events, it</p><p>follows that the probability that a random variable takes on a value in an</p><p>interval of finite length is the “sum” of probabilities or the integral of fX(x)</p><p>dx over the interval. Thus the area under the PDF in an interval represents</p><p>the probability that the random variable will take on a value in that interval</p><p>images</p><p>The probability that a continuous random variable X takes on a specific</p><p>value x is zero, since the length of the interval has vanished. The value of</p><p>fX(x) is not itself a probability; it is only the measure of the density or</p><p>intensity of probability at the point. It follows that fX(x) need not be</p><p>restricted to values less than 1, but two conditions must hold:</p><p>images</p><p>images</p><p>These properties can be verified by inspection for the following example.</p><p>For illustration of the PDF see Fig. 2.2.2a. Here the engineer has called the</p><p>yield stress of a standard tensile specimen of A36 steel a random variable Y.</p><p>He has assigned this variable the triangular probability density function. The</p><p>bases of the assumption of this particular form are observed experimental</p><p>data and the simplicity of its shape. Its range (35 to 55 ksi) and mode (41</p><p>ksi) define a triangular PDF inasmuch as the area must be unity. Although</p><p>simple in shape, this function is somewhat awkward mathematically</p><p>images</p><p>The shaded area between y2 and y3 represents the probability that the yield</p><p>strength will lie in this range, and the</p><p>shaded region from y = 35 to y1 is</p><p>equal in area to the probability that the yield strength is less than y1. For the</p><p>values of y1, y2, and y3 in the ranges shown in Fig. 2.2.2a,</p><p>images</p><p>Continuous random variable and the CDF Again the cumulative</p><p>distribution function, or CDF, is an alternate form by which to describe the</p><p>probability distribution of a random variable. Its definition is unchanged for</p><p>a continuous random variable:</p><p>images</p><p>The right-hand side of this equation may be written P[– ∞ ≤ X ≤ x] and thus,</p><p>for continuous random variables [by Eq. (2.2.5)],</p><p>images</p><p>where u has been used as the dummy variable of integration to avoid</p><p>confusion with the limit of integration x [the argument of the function</p><p>FX(x)]. The CDF of the steel yield stress random variable is shown in Fig.</p><p>2.2.2b.</p><p>images</p><p>Fig. 2.2.2 Steel-yield-stress illustration, (a) Probability density function; (b)</p><p>cumulative distribution function.</p><p>In addition, the PDF can be determined if the CDF is known, since fX(x) is</p><p>simply the slope or derivative of FX(x):</p><p>images</p><p>It is sometimes desirable to use as models mixed random variables, which</p><p>are a combination of the continuous and discrete variety. In this case one can</p><p>always define a meaningful (discontinuous) CDF, but its derivative, a PDF,</p><p>cannot be found without resort to such artifices as Dirac delta functions.†</p><p>The mixed random variable will seldom be considered explicitly in this</p><p>work, since an understanding of the discrete and continuous variables is</p><p>sufficient to permit the reader to deal with this hybrid form. One use is</p><p>pictured in Fig. 2.2.5.</p><p>The cumulative distribution function of any type of random variable—</p><p>discrete, continuous, or mixed—has certain easily verified properties which</p><p>follow from its definition and from the properties of probabilities:‡</p><p>images</p><p>images</p><p>images</p><p>images</p><p>images</p><p>Equation 2.2.14 implies that the CDF is a function which is monotonic and</p><p>nondecreasing (it may be flat in some regions). Cumulative distribution</p><p>functions for discrete, continuous, and mixed random variables are</p><p>illustrated in Figs. 2.2.3 to 2.2.5.</p><p>Histograms and probability distribution models Although they may be</p><p>similar in appearance, the distinction between histograms (Chap. 1) and</p><p>density functions (Chap. 2) and the distinction between cumulative</p><p>frequency polygons and cumulative distribution functions must be well</p><p>understood. The figures presented in Chap. 1 are representations of observed</p><p>empirical data; the functions defined here are descriptions of the probability</p><p>laws of mathematical variables.</p><p>The histogram in Fig. 2.2.6, for example, might represent the observed</p><p>annual runoff data from the watershed of a particular stream. [The first bar</p><p>includes six observations (years) in which the runoff was so small as not to</p><p>be measurable.] In constructing a mathematical model of a river basin, the</p><p>engineer would have use for the stream information represented here.</p><p>Letting the random variable X represent the annual runoff of this area, the</p><p>engineer can construct any number of plausible mathematical models of this</p><p>phenomenon. In particular, any one of the probability laws pictured in Figs.</p><p>2.2.3 to 2.2.5 might be adopted.</p><p>images</p><p>Fig. 2.2.3 Discrete random variable model reproducing histogram of Fig.</p><p>2.2.6. (a) Probability mass function; (b) cumulative distribution function.</p><p>images</p><p>Fig. 2.2.4 Continuous random variable approximately modeling histogram</p><p>of Fig. 2.2.6. (a) Probability density function; (b) cumulative distribution</p><p>function.</p><p>The first model, Fig. 2.2.3, within the restriction of a discrete random</p><p>variable, reproduces exactly the frequencies reported in the histogram. The</p><p>engineer may have no reason to alter his assigned probabilities from the</p><p>observed relative frequencies, even though another sequence of observations</p><p>would change these frequencies to some degree. The second model, Fig.</p><p>2.2.4, enjoys the computational convenience frequently associated with</p><p>continuous random variables. In assigning probabilities to the mathematical</p><p>model, the observed frequencies have been smoothed to a series of straight</p><p>lines to facilitate their description and use. A third possible model, Fig.</p><p>2.2.5, employs, like the second model, a continuous random variable to</p><p>describe the continuous spectrum of physically possible values of runoff, but</p><p>also accounts explicitly for the important possibility that the runoff is exactly</p><p>zero. Over the continuous range the density function has been given a</p><p>smooth, easily described curve whose general mathematical form may be</p><p>determined by arguments† about the physical process leading to the runoff.</p><p>Unfortunately it is not possible to state in general which is the “best”</p><p>mathematical model of the physical phenomenon. The questions of</p><p>constructing models and the relationship of observed data to such models</p><p>will be discussed throughout this work. Commonly used models will be</p><p>discussed in Chap. 3. The use of data to estimate the parameters of the</p><p>model will be considered in Chap. 4, where we also discuss techniques for</p><p>choosing and evaluating models when sufficient data is available.</p><p>images</p><p>Fig. 2.2.5 Mixed random variable approximately modeling characteristics of</p><p>histogram of Fig. 2.2.6. (a) Graphical description of a mixed probability</p><p>law; (b) cumulative distribution function.</p><p>images</p><p>Fig. 2.2.6 Histogram of observed annual runoff (60-year record).</p><p>Often distributions of interest are developed from assumptions about</p><p>underlying components or “mechanisms” of the phenomenon. Two</p><p>examples follow. A third illustration draws the PDF from a comparison with</p><p>data.</p><p>Illustration: Load location An engineer concerned with the forces caused</p><p>by arbitrarily located concentrated loads on floor systems might be</p><p>interested in the distribution of the distance X from the load to the nearest</p><p>edge support. He assumes that the load will be located “at random,”</p><p>implying here that the probability that the load lies in any region of the floor</p><p>is proportional only to the area of that region. He is considering a square bay</p><p>2a by 2a in size.</p><p>From this assumption about the location of the load, we can conclude</p><p>(see Fig. 2.2.7) that</p><p>images</p><p>images</p><p>Fig. 2.2.7 Floor-system illustration. Shaded region is area in which distance</p><p>to nearest edge is equal to or less than x.</p><p>The density function is thus</p><p>images</p><p>which is simply a triangle. That this is a proper probability distribution</p><p>is verified by noting that at x = 0, FX(0) = 0, and at x = a, FX(a) = 1.</p><p>Illustration: Quality control Specification limits on materials (e.g.,</p><p>concrete, asphalt, soil, etc.) are often written recognizing that there is a</p><p>small, “acceptable” probability p that an individual specimen will fail to</p><p>meet the limit even though the batch is satisfactory. As a result more than</p><p>one specimen may be called for when controlling the quality of the material.</p><p>What is the probability mass function of N, the number of specimens which</p><p>will fail to meet the specifications in a sample of size three when the</p><p>material is satisfactory? The probability that any specimen is unsatisfactory</p><p>is p.</p><p>Assuming independence of the specimens,</p><p>images</p><p>This last expression follows from the fact that any one of the three</p><p>sequences {s,s,u}, {s,u,s}, {u,s,s} (where s indicates a satisfactory specimen</p><p>and u an unsatisfactory one) will lead to a value of N equal to 1, and each</p><p>sequence has probability of occurrence p(1 – p)2. Similarly,</p><p>images</p><p>These four terms can be expressed as a function:</p><p>images</p><p>This function is plotted in Fig. 2.2.8 for several values of the parameter</p><p>p. (We will study distributions of this general form in Sec. 3.1.2.)</p><p>That the PMF is proper for any value of p can be verified by</p><p>expanding the individual terms and adding them together by like powers of</p><p>p. The sum is unity.</p><p>In quality-control practice, of course, the engineer must make a decision</p><p>based on an observation of, say, two bad specimens in a sample size</p><p>of three</p><p>about whether the material meets specifications or not. If, under the</p><p>assumption that the material is satisfactory, the likelihood of such an event</p><p>is, in fact, calculated to be very small, the engineer will usually decide (i.e.,</p><p>act as if) the material is not satisfactory.</p><p>Illustration: Annual maximum wind velocity A structural engineer is</p><p>interested in the design of a tall tower for wind loads. He obtains data for a</p><p>number of years of the maximum annual wind velocity near the site and</p><p>finds that when a histogram of the data is plotted, it is satisfactorily modeled</p><p>from a probability viewpoint by a continuous probability distribution of the</p><p>negative exponential form.† If X is maximum annual wind velocity, the PDF</p><p>of X is of the form:</p><p>images</p><p>Fig. 2.2.8 Quality-control illustration. Plots of</p><p>images</p><p>images</p><p>where k is a constant which can be found by recognizing that the</p><p>integral of fX(x) over 0 to ∞ must equal unity. Hence</p><p>images</p><p>or</p><p>images</p><p>yielding</p><p>images</p><p>The CDF is found by integration:</p><p>images</p><p>The record shows that the probability of maximum annual wind</p><p>velocities less than 70 mph is approximately 0.9. This estimate affords an</p><p>estimate of the parameter λ. (Other methods for parameter estimation are</p><p>discussed in Chap. 4.)</p><p>images</p><p>Then,</p><p>images</p><p>images</p><p>The PDF and CDF of X are shown in Fig. 2.2.9.</p><p>One minus FX(x) is important because design decisions are based on</p><p>the probability of large wind velocities. Define † the complementary</p><p>distribution function as</p><p>images</p><p>GX(x) is the probability of finding the maximum wind velocity in</p><p>any year greater than x. The probability of a maximum annual wind velocity</p><p>between 35 and 70 mph is indicated on the PDF of Fig. 2.2.9 along with the</p><p>probability of a maximum annual wind velocity equal to or greater than 140</p><p>mph. Equation (2.2.23) can be used to determine numerical values of these</p><p>probabilities. Their use in engineering design will be discussed in Secs. 3.1</p><p>and 3.3.3.</p><p>2.2.2 Jointly Distributed Random Variables</p><p>In Sec. 2.1 we discussed examples, such as counting cars and counting</p><p>trucks, in which two-dimensional sample spaces are involved. When two or</p><p>more random variables are being considered simultaneously, their joint</p><p>behavior is determined by a joint probability law, which can in turn be</p><p>described by a joint cumulative distribution function. Also, if both random</p><p>variables are discrete, a joint probability mass function can be used to</p><p>describe their governing law, and if both variables are continuous, a joint</p><p>probability density function is applicable. Mixed joint distributions are also</p><p>encountered in practice, but they require no new techniques.</p><p>images</p><p>Fig. 2.2.9 Wind-velocity illustration, (a) PDF of X and (b) CDF of X (X is</p><p>maximum annual wind velocity).</p><p>Joint PMF The joint probability mass function pX,Y(x,y) of two discrete</p><p>random variables X and Y is defined as</p><p>images</p><p>It can be plotted in a three-dimensional form analogous to the two-</p><p>dimensional PMF of a single random variable. The joint cumulative</p><p>distribution function is defined as</p><p>images</p><p>Consider an example of two discrete variables whose joint behavior must be</p><p>dealt with. One, X, is the random number of vehicles passing a point in a 30-</p><p>sec time interval. Variability in traffic flow is the cause of delays and</p><p>congestion. The other random variable is Y, the number of vehicles in the</p><p>same 30-sec interval actually recorded by a particular, imperfect traffic</p><p>counter. This device responds to pressure on a cable placed across one or</p><p>more traffic lanes, and it records the total number of such pressure</p><p>applications during successive 30-sec intervals. It is used by the traffic</p><p>engineer to estimate the number of vehicles which have used the road during</p><p>given time intervals. Owing, however, to dynamic effects (causing wheels to</p><p>be off the ground) and to mechanical inability of the counter to respond to</p><p>all pulses, the actual number of vehicles X and the recorded number of</p><p>vehicles Y are not always in agreement. Data were gathered in order to</p><p>determine the nature and magnitude of this lack of reliability in the counter.</p><p>Simultaneous observations of X and Y in many different 30-sec intervals led</p><p>to a scattergram (Sec. 1.3). The engineer adopted directly the observed</p><p>relative frequencies as probability assignments in his mathematical model,†</p><p>yielding the joint probability mass function graphed in Fig. 2.2.10. Notice</p><p>by the strong probability masses on the diagonal (0, 0; 1, 1; etc.) that the</p><p>counter is usually correct. Note too that it is not possible for Y to take on a</p><p>value greater than X in this example. The joint CDF could also be plotted,</p><p>appearing always something like an exterior corner of an irregular staircase,</p><p>but it seldom proves useful to do so.</p><p>images</p><p>Fig. 2.2.10 Joint PMF of actual traffic and traffic recorded by counter.</p><p>The probability of any event of interest is found by determining the pairs of</p><p>values of X and Y which lead to this event and then summing over all such</p><p>pairs. For example, the probability of C, the event that an arbitrary count is</p><p>not in error, is</p><p>images</p><p>The probability of an error by the counter is</p><p>images</p><p>This probability is most easily calculated as</p><p>images</p><p>Since, to be properly defined,</p><p>images</p><p>Marginal PMF A number of functions related to the joint PMF are of value.</p><p>The behavior of a particular variable irrespective of the other is described by</p><p>the marginal PMF. It is found by summing over all values of the disregarded</p><p>variable. Formally,</p><p>images</p><p>images</p><p>Similar expressions hold for the marginal distribution of Y.</p><p>images</p><p>Fig. 2.2.11 (a) Marginal PMF of actual traffic X and (b) marginal PMF</p><p>counter response Y.</p><p>In the example here the distribution of X, the actual number of cars, is found</p><p>for each value x by summing all spikes in the y direction. For example,</p><p>images</p><p>The marginal distributions of X and Y are plotted on Fig. 2.2.11. It should be</p><p>pointed out that generally the marginal distributions are not sufficient to</p><p>specify the joint behavior of the random variables. In this example, the joint</p><p>PMF requires specification of (5)(5) = 25 numbers, while the two marginal</p><p>distributions contain only 5 + 5 = 10 pieces of information. The conditions</p><p>under which the marginals are sufficient to define the joint will be discussed</p><p>shortly.</p><p>Conditional PMF A second type of distribution which can be obtained from</p><p>the joint distribution is also of interest. If the value of one of the variables is</p><p>known, say Y = y0, the relative likelihoods of the various values of the other</p><p>variable are given by pX,Y(x,y0). If these values are renormalized so that</p><p>their sum is unity, they will form a proper distribution function. This</p><p>distribution is called the conditional probability mass function of X given Y,</p><p>PX|Y(x,y). The normalization is performed by dividing each of the values by</p><p>their sum. For Y given equal to any particular value y,</p><p>images</p><p>(The function is undefined if pY(y) equals zero.) Notice that the denominator</p><p>is simply the marginal distribution of Y evaluated at the given value of Y.</p><p>The conditional PMF of X is a proper distribution function in x, that is,</p><p>images</p><p>and</p><p>images</p><p>The conditional distribution of Y given X is, of course, defined in a</p><p>symmetrical way.</p><p>It is the relationship between the conditional distribution and the marginal</p><p>distribution that determines how much an observation of one variable helps</p><p>in the prediction of the other. In the traffic-counter example, our interest is in</p><p>X, the actual number of cars which have passed in a particular interval. The</p><p>marginal distribution of X is initially our best statement regarding the</p><p>relative likelihoods of the various possible values of X. An observation of Y</p><p>(the mechanically recorded number) should, however, alter these</p><p>likelihoods. Suppose the counter reads Y = 1 in a particular interval. The</p><p>actual number of cars is not known with certainty, but the relative</p><p>likelihoods</p><p>of different values are now given by PX,Y(x,1) or 0, 0.36, 0.03,</p><p>0.01, and 0 for x equal, to 0, 1, 2, 3, and 4, respectively. Normalized by their</p><p>sum, these likelihoods become the conditional distribution PX|Y(x,1), which</p><p>is plotted in Fig. 2.2.12. As expected, the probability is now high that X</p><p>equals 1, but the imperfect nature of the counter does not permit this</p><p>statement to be made with absolute certainty. The better the counter, the</p><p>more peaked it will make this conditional distribution relative to the</p><p>marginal distribution. In the words of Sec. 2.1, the more closely the</p><p>measured quantity (here Y) is related or correlated with the quantity of</p><p>interest (here X), the more “sharply” can we predict X given a measured</p><p>value F.</p><p>Other conditional distributions It is of course true that any number of</p><p>potentially useful conditional distributions can be constructed from a joint</p><p>probability law depending on the event conditioned on. Thus one might seek</p><p>the distribution of X given that Y is greater than y:</p><p>images</p><p>Fig. 2.2.12 Conditional PMF of traffic given that the counter reads unity.</p><p>images</p><p>or the conditional distribution of Y given that X was in a particular interval:</p><p>images</p><p>We will encounter examples of such conditional distributions at various</p><p>points in this text. Their treatment and meaning, however, is obvious once</p><p>the notation is defined. In this notational form the fundamental conditional</p><p>PMF [Eq. (2.2.29)] becomes</p><p>images</p><p>Joint PMF from marginal and conditional probabilities As mentioned</p><p>before with regard to events, conditional probabilities are often more readily</p><p>determined in practice than are the joint probabilities. Together with one of</p><p>the marginal PMF’s a conditional PMF can be used to compute the joint</p><p>PMF, since, from Eq. (2.2.29),</p><p>images</p><p>Illustration: Analytical approach to traffic-counter model We shall</p><p>illustrate the use of the previous equation by showing that the joint</p><p>distribution in the traffic-counter example might have been determined</p><p>without data by constructing a mathematical model of the probabilistic</p><p>mechanism generating the randomness. A commonly used model of traffic</p><p>flow (to be discussed in Sec. 3.2.1) suggests that the number of cars passing</p><p>a fixed point in a given interval of time has a discrete distribution of the</p><p>mathematical form</p><p>images</p><p>where v is the average number of cars in all such intervals. If it is</p><p>assumed, too, that each vehicle is recorded only with probability p, then in</p><p>Sec. 3.1.2 we shall learn that the conditional distribution of Y given X = x</p><p>must be</p><p>images</p><p>[The argument is a generalization of that which led to Eq. (2.2.18).]</p><p>Hence we need data or information sufficient only to estimate v, which is</p><p>related to the average flow, and p, the unreliability of the counter, rather than</p><p>all the values in the joint PMF. This is true because the joint PMF follows</p><p>from the marginal and conditional above:</p><p>images</p><p>Without being able to argue through the conditional PMF, it would</p><p>have been extremely difficult for the engineer to derive this complicated</p><p>joint PMF directly. For v = 1.3 cars and p = 0.9, its shape is very nearly that</p><p>given in Fig. 2.2.11, although the sample space now extends beyond 4 to</p><p>infinity. It is not expected or important at this stage that the reader absorb the</p><p>details of this model. It is brought up here to illustrate the utility of the</p><p>conditional distribution and to display the “more mathematical” form of</p><p>some commonly encountered discrete distributions. It points out that, if the</p><p>physical process is well understood, the engineer may attempt to construct a</p><p>reasonable model of the generating mechanism involving only a few</p><p>parameters (here v and p), rather than rely on observed data to provide</p><p>estimates of all the probability values of the distribution function.</p><p>Joint PDF and CDF The functions associated with jointly distributed</p><p>continuous random variables are totally analogous with those of discrete</p><p>variables, but with density functions replacing mass functions. Using the</p><p>same type of argument employed in Sec. 2.2.1, the probability that X lies in</p><p>the interval {x, x + dx} and Y lies in the interval {y, y + dy} is fX,Y(x,y) dx dy.</p><p>This function fX,Y(x,y) is called the joint probability density function.</p><p>The probability of the joint occurrence of X and Y in some region in the</p><p>sample space is determined by integration of the joint PDF over that region.</p><p>For example,</p><p>images</p><p>This is simply the volume under the function fX,Y(x,y) over the region. If the</p><p>region is not a simple rectangle, the integral may become more difficult to</p><p>evaluate, but the problem is no longer one of probability but calculus (and</p><p>all its techniques to simplify integration, such as change of variables, are</p><p>appropriate).</p><p>Clearly the joint PDF must satisfy the conditions</p><p>images</p><p>images</p><p>The joint cumulative distribution function is defined as before and can be</p><p>computed from the density function by applying Eq. (2.2.38). Thus</p><p>images</p><p>in which dummy variables of integration, x0 and y0, have been used to</p><p>emphasize that the arguments of the CDF, x and y, appear in the limits of the</p><p>integral. The properties of the function are analogous to those given in Eqs.</p><p>(2.2.11) to (2.2.15) for the CDF of a single variable.</p><p>It should not be unexpected that, as in Sec. 2.2.1, the density function is a</p><p>derivative of the cumulative function, now a partial derivative,</p><p>images</p><p>Marginal PDF As with the discrete random variables, one frequently has</p><p>need for marginal distributions. To eliminate consideration of Y in studying</p><p>the behavior of X, one need only integrate the joint density function over all</p><p>values of Y and determine the marginal PDF of X, fX(x):</p><p>images</p><p>The marginal cumulative distribution function of X, FX(x), is, consequently,</p><p>images</p><p>or</p><p>images</p><p>which implies</p><p>images</p><p>Symmetrical results hold for the marginal distribution of Y.</p><p>Conditional PDF If one is given the value of one variable, Y = y0, say, the</p><p>relative likelihood of X taking a value in the interval x, x + dx is fX,Y(x,y0)</p><p>dx. To yield a proper density function (i.e., one whose integral over all</p><p>values of x is unity) these values must be renormalized by dividing them by</p><p>their sum, which is</p><p>images</p><p>In this manner we are led, plausibly if not rigorously,† to the definition of</p><p>the conditional PDF of X given Y as</p><p>images</p><p>The conditional cumulative distribution is</p><p>images</p><p>As with discrete variables, distributions based on other conditioning events,</p><p>for example, images may also prove useful. Their definitions and</p><p>interpretations should be obvious.</p><p>Sketches of some joint PDF’s appear in Fig. 2.2.13 in the form of contours</p><p>of equal values. Graphical interpretations of marginal and conditional</p><p>density functions also appear there.</p><p>images</p><p>Fig. 2.2.13 Three types of joint PDF’s illustrating the relationships between</p><p>the two random variables, (a) Contours (higher values of X usually occur</p><p>with lower values of Y); (b) marginal PDF’s (higher values of X occur about</p><p>equally with higher and lower values of Y); (c) cross sections at y = y0 and y</p><p>= y1 (higher values of X usually occur with higher values of Y).</p><p>As an illustration of joint continuous random variables, consider the flows X</p><p>and Y in two different streams in the same day. Interest in their joint</p><p>probabilistic behavior might arise because the streams feed the same</p><p>reservoir. If, as in the case in Fig. 2.2.13c, high flows in one stream are</p><p>likely to occur simultaneously with high flows in the other, their joint</p><p>influence on the reservoir will not be the same as it would if this were not</p><p>the case. Assume that the joint distribution in this simplified illustration is</p><p>that shown in Fig. 2.2.14. By inspection, its equation is</p><p>images</p><p>Fig. 2.2.14 Joint PDF of two stream flows.</p><p>images</p><p>The constant, C = 2.5 × 10–7, was evaluated after the shape was fixed so that</p><p>the integral over the entire sample space would be unity.</p><p>The probability of such events as “the flow X is more than twice as great as</p><p>the flow Y” can be found</p><p>by integration over the proper region. The portion</p><p>of the sample space where this is true is shown shaded in Fig. 2.2.15a. The</p><p>probability of this event is the volume under the PDF over this region, Fig.</p><p>2.2.15b. Formally,</p><p>images</p><p>By carrying out the integrations or by inspection of the volume, knowing it</p><p>to be (⅓) (area of base) (height):</p><p>images</p><p>The marginal distributions of X and Y are formally</p><p>images</p><p>and</p><p>images</p><p>These functions are plotted in Fig. 2.2.16. Their shapes could have been</p><p>anticipated by inspection of the joint PDF.</p><p>Without further information, prediction of the flow X must be based on fX(x).</p><p>It might be, however, that the engineer has in mind using an observation of Y</p><p>(the flow in the “smaller” stream) to predict X more “accurately,” just as the</p><p>traffic engineer used an inexpensive, but unreliable, mechanical counter to</p><p>provide a sharper knowledge of the number of vehicles which passed. As we</p><p>have seen, the relationship between the conditional distribution function and</p><p>the marginal distribution provides a measure of the information about X</p><p>added by the knowledge of F. The conditional PDF of X given Y is</p><p>images</p><p>Fig. 2.2.15 Stream-flow illustration, (a) Sample space showing the event X ≥</p><p>2Y; (b) volume equal to P[X ≥ 2Y]</p><p>images</p><p>Fig. 2.2.16 Marginal distributions of stream flow from joint PDF of Fig.</p><p>2.2.14. (a) Marginal PDF of X; (b) Marginal PDF of Y.</p><p>images</p><p>Notice that this is unchanged from the marginal distribution fX(x). The</p><p>knowledge of Y has not altered the distribution of X and hence has not</p><p>provided the engineer any new information in his quest to predict or</p><p>describe the behavior of X. This case is an example of a very important</p><p>notion.</p><p>Independent random variables In general, if the conditional distribution</p><p>fX|Y(x,y) is identical to the marginal distribution fX(x), X and Y are said to be</p><p>(stochastically) independent random variables. Similarly, for discrete</p><p>variables, if, for all values of y</p><p>images</p><p>then X and Y are independent. The notion of independence of random</p><p>variables is directly parallel to that of independence of events, for recall that</p><p>Eq. (2.2.46) can be written in terms of probabilities of events as</p><p>images</p><p>If the random variables X and Y are independent, events related to X are</p><p>independent of those related to Y. As a result of the definition above, the</p><p>following statements hold if X and Y are independent and continuous</p><p>(analogous relationships for discrete or mixed random variables can be</p><p>derived):</p><p>images</p><p>images</p><p>images</p><p>images</p><p>images</p><p>As with events, in engineering practice independence of two or more</p><p>random variables is usually a property attributed to them by the engineer</p><p>because he thinks they are unrelated. This assumption permits him to</p><p>determine joint distribution functions from only the marginals [Eq. (2.2.50)].</p><p>In general this is not possible, and both a marginal and a conditional are</p><p>required.</p><p>The concept of probabilistic independence is central to the successful</p><p>application of probability theory. From a purely practical point of view, the</p><p>analysis of many probabilistic models would become hopelessly complex if</p><p>the engineer were unwilling to adopt the assumption of independence of</p><p>certain random variables in a number of key situations. Many examples of</p><p>assumed independence of random variables will be found in this text.</p><p>Three or more random variables Attention has been focused in this</p><p>section on cases involving only two random variables but, at least in theory,</p><p>the extensions of these notions to any number of jointly distributed random</p><p>variables should not be difficult. In fact, however, the calculus—the partial</p><p>differentiation and multiple integrations over bounded regions—may</p><p>become unwieldy. Many new combinations of functions become possible</p><p>when more variables are considered. A few should be mentioned for</p><p>illustration. For example, with only three variables X, Y, Z with joint CDF</p><p>fX,Y,Z(x,y,z) there might be interest in joint marginal CDF’s, such as</p><p>images</p><p>as well as in simple marginal CDF’s, such as</p><p>images</p><p>Joint probability density functions are, as above, partial derivatives:†</p><p>images</p><p>images</p><p>images</p><p>Joint conditional PDF’s are also now possible, and they are defined as might</p><p>be expected:</p><p>images</p><p>The simple conditional PDF follows this pattern:</p><p>images</p><p>If the random variables X, Y, and Z are mutually independent,‡</p><p>images</p><p>and conditional distributions reduce to marginal or joint PDF’s. For</p><p>example,</p><p>images</p><p>Illustration: Reliability of a system subjected to n random demands In a</p><p>capacity-demand situation such as those discussed in Sec. 2.1, it is often the</p><p>case that the system is subjected to a succession of n demands (annual</p><p>maximum flows, or extreme winds, for example). Assuming that these</p><p>random variables D1, D2, . . ., Dn are independent and identically</p><p>distributed, and that the random capacity C is independent of all these</p><p>demands, we can find the probability of the “failure” event, A = [at least one</p><p>of the n demands exceeds the capacity], as follows. Assume the random</p><p>variables are all discrete.</p><p>images</p><p>Expanding, using Eq. (2.1.12),</p><p>images</p><p>Since capacity C is assumed independent of the Di,</p><p>images</p><p>Since the Di are assumed mutually independent,</p><p>images</p><p>and since the Di are assumed identically distributed (say, all distributed</p><p>like a random variable denoted simply D),</p><p>images</p><p>Thus</p><p>images</p><p>For example, if the PMF’s of D and C have values at the relative</p><p>positions shown in Fig. 2.2.17, then</p><p>images</p><p>Assuming that np1 and images are small compared with unity,</p><p>images</p><p>Assuming that images is small compared with 1, images is small</p><p>compared to images</p><p>images</p><p>Depending on the magnitudes of p1p2, images the last term may or</p><p>may not be negligible.</p><p>The relative magnitudes and locations of the probability spikes in</p><p>Fig. 2.2.17 are intended to be suggestive of the common design situation</p><p>where c3 is the “intended” capacity expected by the designer and images is</p><p>the anticipated or typical “design” demand. Failure of the system will occur</p><p>during the design lifetime of n demands if any of the following happens:</p><p>1. A demand images rare in magnitude or unanticipated in kind,</p><p>occurs at some time during the lifetime.</p><p>images</p><p>Fig. 2.2.17 Distributions of demand and capacity. Positions of spikes on c</p><p>axis are relative only.</p><p>2. An unusually low capacity c1 is obtained either through construction</p><p>inadequacy or through a gross misjudgement by the engineer as to the</p><p>design capacity which would result from his specified design.</p><p>3. There exists a combination or joint occurrence of a moderately</p><p>higher demand images and a capacity of the only moderately lower value</p><p>c2.</p><p>After simplification and approximation these three events make their</p><p>appearance as the three terms in Eq. (2.2.69).</p><p>Notice that in general the probability of a system failure increases</p><p>with design life n; a system intended to last longer is exposed to a greater</p><p>risk that one of the higher load values occurs in its lifetime. One term,</p><p>images, does not grow with n, however, since, if this very low capacity is</p><p>obtained, the system will fail with certainty under the first demand. If the</p><p>capacity is not this low, further demands do not increase the likelihood of</p><p>failure due to such a source.</p><p>2.3 DERIVED DISTRIBUTIONS</p><p>Much of science and engineering is based on functional relationships which</p><p>predict the value of one (dependent) variable given any value of another</p><p>(independent) variable. Static pressure is a function of fluid density; yield</p><p>force, a function of cross-sectional area; and so forth. If, in a probabilistic</p><p>formulation, the independent variable is considered to be a random variable,</p><p>this randomness is imparted to those variables which are functionally</p><p>dependent upon it. The purpose of this section is to develop methods of</p><p>determining the probability law of functionally</p><p>The necessity of making this new material accessible to engineers who</p><p>have already been trained in classical statistics was the source of what the</p><p>authors feel to be their major compromise. The authors are frankly</p><p>Bayesianists, but because these classically trained engineers are making the</p><p>present developments in the field and because they are the engineering</p><p>educators of the future, we decided to present classical as well as modern</p><p>statistics. It is hoped that a reader can use the book either way, if not both,</p><p>and that this approach will accelerate the development and spread of</p><p>probabilistic methods within our profession. Inclusion of both approaches</p><p>will also aid those with some previous experience to interpret and judge the</p><p>methods he is more familiar with. Finally, knowledge of both approaches is</p><p>desirable, if only to understand the meaning and the limitations of the</p><p>widely used classical methods. We apologize for the fact that the inclusion</p><p>of both methods has led to a certain lack of uniformity in position and in</p><p>treatment that the reader may find puzzling. We wish to thank the many</p><p>students and colleagues who have contributed over the past seven years to</p><p>the development of this text.</p><p>The book can be used in a variety of ways. The reader seeking no more</p><p>than a brief elementary introduction to probability might read as little as</p><p>Chap. 1, Secs. 2.1, 2.2.1, 2.4.1, 3.1.1, 3.1.2, 3.2.1, 3.2.2, 3.3, and 4.1.1.</p><p>Minimum course coverage should probably include, in addition, the</p><p>material in Secs. 2.2.2, 2.3.1, 2.4.2, 3.1.3, 3.6.2, 4.1.2, 4.1.3, 4.2, 4.4, and</p><p>Chap. 5. (Much of the material in Secs. 3.1, 3.2, and 3.4 can be covered in</p><p>lecture courses as illustrations of the material in Chap. 2 rather than</p><p>separately.) At Stanford and M.I.T. we teach several quite different courses,</p><p>using the same text and relying on it to supplement the material not</p><p>emphasized in lectures.</p><p>JACK R. BENJAMIN</p><p>C. ALLIN CORNELL</p><p>Contents</p><p>Preface</p><p>Introduction</p><p>Chapter 1 Data Reduction</p><p>1.1 Graphical Displays</p><p>1.2 Numerical Summaries</p><p>1.3 Data Observed in Pairs</p><p>1.4 Summary for Chapter 1</p><p>Chapter 2 Elements of Probability Theory</p><p>2.1 Random Events</p><p>2.1.1 Sample Space and Events</p><p>2.1.2 Probability Measure</p><p>2.1.3 Simple Probabilities of Events</p><p>2.1.4 Summary</p><p>2.2 Random Variables and Distributions</p><p>2.2.1 Random Variables</p><p>2.2.2 Jointly Distributed Random Variables</p><p>2.3 Derived Distributions</p><p>2.3.1 One-variable Transformations: Y = g(X)</p><p>2.3.2 Functions of Two Random Variables</p><p>2.3.3 Elementary Simulation</p><p>2.3.4 Summary</p><p>2.4 Moments and Expectation</p><p>2.4.1 Moments of a Random Variable</p><p>2.4.2 Expectation of a Function of a Random Variable</p><p>2.4.3 Expectation and Jointly Distributed Random Variables</p><p>2.4.4 Approximate Moments and Distributions of Functions</p><p>2.4.5 Summary</p><p>2.5 Summary for Chapter 2</p><p>Chapter 3 Common Probabilistic Models</p><p>3.1 Models from Simple Discrete Random Trials</p><p>3.1.1 A Single Trial: The Bernoulli Distribution</p><p>3.1.2 Repeated Trials: The Binomial Distribution</p><p>3.1.3 Repeated Trials: The Geometric and Negative</p><p>Binomial Distributions</p><p>3.1.4 Summary</p><p>3.2 Models from Random Occurrences</p><p>3.2.1 Counting Events: The Poisson Distribution</p><p>3.2.2 Time between Events: The Exponential Distribution</p><p>3.2.3 Time to the kth Event: The Gamma Distribution</p><p>3.2.4 Summary</p><p>3.3 Models from Limiting Cases</p><p>3.3.1 The Model of Sums: The Normal Distribution</p><p>3.3.2 The Model of Products: The Lognormal Distribution</p><p>3.3.3 The Model of Extremes: The Extreme Value</p><p>Distributions</p><p>3.3.4 Summary</p><p>3.4 Additional Common Distributions</p><p>3.4.1 The Equally Likely Model: The Rectangular or</p><p>Uniform Distribution</p><p>3.4.2 The Beta Distribution</p><p>3.4.3 Some Normal Related Distributions: Chi-square, Chi,</p><p>t, and F</p><p>3.4.4 Summary</p><p>3.5 Modified Distributions</p><p>3.5.1 Shifted and Transformed Distributions</p><p>3.5.2 Truncated and Censored Distributions</p><p>3.5.3 Compound Distributions</p><p>3.5.4 Summary</p><p>3.6 Multivariate Models</p><p>3.6.1 Counting Multiple Events: The Multinomial</p><p>Distribution</p><p>3.6.2 The Multivariate Normal Distribution</p><p>3.6.3 Summary</p><p>3.7 Markov Chains</p><p>3.7.1 Simple Markov Chains</p><p>3.7.2 Two-state Homogeneous Chains</p><p>3.7.3 Multistate Markov Chains</p><p>3.7.4 Summary</p><p>3.8 Summary for Chapter 3</p><p>Chapter 4 Probabilistic Models and Observed Data</p><p>4.1 Estimation of Model Parameters</p><p>4.1.1 The Method of Moments</p><p>4.1.2 The Properties of Estimators: Their First- and Second-</p><p>order Moments</p><p>4.1.3 The Distributions of Estimators and Confidence-</p><p>interval Estimation</p><p>4.1.4 The Method of Maximum Likelihood</p><p>4.1.5 Summary</p><p>4.2 Significance Testing</p><p>4.2.1 Hypothesis Testing</p><p>4.2.2 Some Common Hypothesis Tests</p><p>4.2.3 Summary</p><p>4.3 Statistical Analysis of Linear Models</p><p>4.3.1 Linear Models</p><p>4.3.2 Statistical Analysis of Simple Linear Models</p><p>4.3.3 Summary</p><p>4.4 Model Verification</p><p>4.4.1 Comparing Shapes: Histograms and Probability Paper</p><p>4.4.2 “Goodness-of-fit” Significance Tests</p><p>4.4.3 Summary</p><p>4.5 Empirical Selection of Models</p><p>4.5.1 Model Selection: Illustration I, Loading Times</p><p>4.5.2 Model Selection: Illustration II, Maximum Annual</p><p>Flows</p><p>4.5.3 Summary</p><p>4.6 Summary of Chapter 4</p><p>Chapter 5 Elementary Bayesian Decision Theory</p><p>5.1 Decisions with Given Information</p><p>5.1.1 The Decision Model</p><p>5.1.2 Expected-value Decisions</p><p>5.1.3 Probability Assignments</p><p>5.1.4 Analysis of the Decision Tree with Given Information</p><p>5.1.5 Summary</p><p>5.2 Terminal Analysis</p><p>5.2.1 Decision Analysis Given New Information</p><p>5.2.2 Summary</p><p>5.3 Preposterior Analysis</p><p>5.3.1 The Complete Decision Model</p><p>5.3.2 Summary</p><p>5.4 Summary for Chapter 5</p><p>Chapter 6 Decision Analysis of Independent Random Processes</p><p>6.1 The Model and Its Prior Analysis</p><p>6.1.1 Prior Analysis of the Special Problem u(a, X)</p><p>6.1.2 More General Relationships between the Process and</p><p>the State of Interest</p><p>6.1.3 Summary</p><p>6.2 Terminal Analysis Given Observations of the Process</p><p>6.2.1 The General Case</p><p>6.2.2 Data-based Decisions: Diffuse Priors</p><p>6.2.3 Use of Conjugate Priors</p><p>6.2.4 Summary</p><p>6.3 The Bayesian Distribution of a Random Variable</p><p>6.3.1 The Simple Case; X Only</p><p>6.3.2 The General Case Y = h(X1 X2, . . ., Xβ)</p><p>6.3.3 Summary</p><p>6.4 Summary of Chapter</p><p>Appendix A Tables</p><p>Table A.1 Values of the Standardized Normal Distribution</p><p>Table A.2 Tables for Evaluation of the CDF of the χ2,</p><p>Gamma, and Poisson Distributions</p><p>Table A.3 Cumulative Distribution of Student’s t</p><p>Distribution</p><p>Table A.4 Properties of Some Standardized Beta</p><p>Distributions</p><p>Table A.5 Values of the Standardized Type I Extreme-value</p><p>Distribution</p><p>Table A.6 F Distribution; Value of z such that Fz(z) = 0.95</p><p>Table A.7 Critical Statistic for the Kolmogorov-Smirnov</p><p>Goodness-of-fit Test</p><p>Table A.8 Table of Random Digits</p><p>Appendix B Derivation of the Asymptotic Extreme-value Distribution</p><p>Name Index</p><p>Subject Index</p><p>PROBABILITY, STATISTICS, AND</p><p>DECISION FOR CIVIL ENGINEERS</p><p>Introduction</p><p>There is some degree of uncertainty connected with all phenomena with</p><p>which civil engineers must work. Peak traffic demands, total annual</p><p>rainfalls, and steel yield strengths, for example, will never have exactly the</p><p>same observed values, even under seemingly identical conditions. Since the</p><p>performance of constructed facilities will depend upon the future values of</p><p>such factors, the design engineer must recognize and deal with this</p><p>uncertainty in a realistic and economical manner.</p><p>How the engineer chooses to treat the uncertainty in a given phenomenon</p><p>depends upon the situation. If the degree of variability is small, and if the</p><p>consequences of any variability will not be significant, the engineer may</p><p>choose to ignore it by simply assuming that the variable will be equal to the</p><p>best available estimate. This estimate might be the average of a number of</p><p>past observations. This is typically done, for example, with the elastic</p><p>constants of materials and the physical dimensions of many objects.</p><p>If, on the other hand, uncertainty is significant, the engineer may choose</p><p>to use a “conservative estimate” of the factor. This has often been done, for</p><p>example, in setting “specified minimum” strength properties of materials</p><p>and when selecting design demands for transportation</p><p>dependent random variables</p><p>when the probability law of the independent † variable is known.</p><p>2.3.1 One-variable Transformations:Y = g(X)</p><p>The nature of such problems is best brought out through a simple, discrete</p><p>example.</p><p>Solution by enumeration For several new high-speed ground-</p><p>transportation systems it has been proposed to utilize small, separate</p><p>vehicles that can be dispatched from one station to another, not on a fixed</p><p>schedule, but whenever full. If the number of persons arriving in a specified</p><p>time period at station A who want to go to station B is a random variable X,</p><p>then Y, the number of vehicles dispatched from A to B in that period, is also</p><p>a random variable, functionally related through the vehicles’ capacities to X.</p><p>For example, if each vehicle carries two persons, then Y = 0 if X = 0 or 1; Y</p><p>= 1 if X = 2 or 3; Y = 2 if X = 4 or 5, etc. (assuming that no customer is</p><p>waiting at the beginning of the time period). A graph of this functional</p><p>relationship is shown in Fig. 2.3.1a.</p><p>The probability mass function of Y, pY(y), can be determined from that of X</p><p>by ennumeration of the values of X which lead to a particular value of Y. For</p><p>example,</p><p>images</p><p>Or, in general,</p><p>images</p><p>A numerical example is shown in Fig. 2.3.1b and c.</p><p>In this and in all derived distribution problems, it is even easier to obtain the</p><p>cumulative distribution function FY(y) from that of X. Thus</p><p>images</p><p>Fig. 2.3.1 High-speed transportation-system example, (a) Functional</p><p>relationship between the number of arriving customers X and the number of</p><p>dispatched vehicles Y; (b) PMF of number of arriving customers; (c) PMF of</p><p>number of dispatched vehicles.</p><p>images</p><p>or</p><p>images</p><p>(It should be stated that this dispatching policy may not be a totally</p><p>satisfactory one; it should perhaps be supplemented by a rule that dispatches</p><p>half-full vehicles if the customer on board has been waiting more than a</p><p>certain time.)</p><p>In this section we shall be concerned, then, with deriving the probability law</p><p>of a random variable which is directly or functionally related to another</p><p>random variable whose probability law is known. Owing to their importance</p><p>in practice and in subsequent chapters, we shall concentrate on analytical</p><p>solutions to problems involving continuous random variables. Problems</p><p>involving discrete random variables can always be treated by enumeration as</p><p>demonstrated above. In practical situations it may also prove advantageous</p><p>to compute solutions to continuous problems in this same way, after having</p><p>first approximated continuous random variables by discrete ones (Sec.</p><p>2.2.1).</p><p>One-to-one transformations For a commonly occurring class of problems</p><p>it is possible to develop simple, explicit formulas as follows. If the function</p><p>y = g(x) relating the two random variables is such that it always increases as</p><p>x increases † and is such that there is only a single value of x for each value</p><p>of y and vice versa, ‡ then Y is less than or equal to some value y0 if and</p><p>only if X is less than or equal to some value x0, namely, that value for which</p><p>y0 = g(x0). (See Fig. 2.3.2.)</p><p>images</p><p>Fig. 2.3.2 A monotonically increasing one-to-one function relating Y to X</p><p>images</p><p>Fig. 2.3.3 Derived distributions, construction-cost illustration, C = 10,000 +</p><p>100H. (a) Functional relationship between cost and time; (b) cumulative</p><p>distribution function of H, given, and C, derived; (c) probability density</p><p>function of H, given, and C, derived.</p><p>Suppose, for example, that for purposes of developing a bid, the distribution</p><p>of the total cost C of constructing a retaining wall is desired. Assume that</p><p>this cost is the cost of materials, which can be accurately predicted to be</p><p>$10,000, plus the cost of a labor crew at $100/hr. The number of hours H to</p><p>complete such a job is uncertain. The total cost is functionally related to H:</p><p>images</p><p>The relationship is monotonically increasing and one-to-one (Fig. 2.3.3).</p><p>The cost is less than any particular value c only if the number of hours is</p><p>less than a particular value h. This is the key to solving such problems.</p><p>Generally we can solve y = g(x) for x to find the inverse function† x = g–</p><p>1(y), which gives the value of x corresponding to any particular value of y.</p><p>(See Figs. 2.3.2 and 2.3.3.) For example, if the value of c corresponding to</p><p>any given h is c = g(h) = 10,000 + 100h, the value of h corresponding to any</p><p>given c is h = g–1(c) = (c – 10,000)/100. If, as another example, y = g(x) =</p><p>aebx, then x = g–1(y) = (1/b) ln (y/a), in which ln (u) denotes the natural</p><p>logarithm of u.</p><p>Under these conditions we can solve directly for the CDF of the dependent</p><p>variable Y, since the probability that Y is less than any value of y is simply</p><p>the probability that X is less than the corresponding value of x, that is, x = g–</p><p>1(y). This probability can be obtained from the known CDF of X:</p><p>images</p><p>or</p><p>images</p><p>Thus, in the construction-cost example,</p><p>images</p><p>Suppose, for example, that the CDF of H is the parabolic function shown in</p><p>Fig. 2.3.3b. Then, in the range of interest, we find FC(c) by simply</p><p>substituting (c – 10,000)/100 for h in FH(h):</p><p>images</p><p>In general, when finding distributions of functionally related random</p><p>variables, we must work with their CDF’s. To obtain a PDF of Y, the CDF</p><p>must be found and then differentiated. A major advantage of the class of</p><p>functional relationships ‡ we are considering, however, is hat we may pass</p><p>directly from the PDF of one to the PDF of the other. Analytically we simply</p><p>need to take the derivative of the CDF, Eq. (2.3.3):</p><p>images</p><p>which can be shown to be</p><p>images</p><p>or, replacing g–1(y) by x, we obtain the more suggestive forms</p><p>images</p><p>or</p><p>images</p><p>In words, the likelihood that Y takes on a value in an interval of width dy</p><p>centered on the value y is equal to the likelihood that X takes on a value in</p><p>an interval centered on the corresponding value x = g–1(y), but of width dx =</p><p>dg–1(y). As shown graphically in Fig. 2.3.4, these interval widths are</p><p>generally not equal, owing to the slope of the function g(x) or g–1(y) at the</p><p>value of y of interest. This slope and hence the ratio of dx to dy may or may</p><p>not be the same for all values of y.</p><p>In our construction-cost example, the ratio dh/dc is constant because the</p><p>relationship between C and H is linear (Fig. 2.3.3a):</p><p>images</p><p>Therefore, given the PDF of H, the PDF of C follows directly from Eq.</p><p>(2.3.4) or Eq. (2.3.5):</p><p>images</p><p>For example, the CDF of H given in Fig. 2.3.3b implies that the PDF of H is</p><p>triangular (Fig. 2.3.3c):</p><p>images</p><p>images</p><p>Fig. 2.3.4 Graphical interpretation of Eqs. (2.3.5) and (2.3.12) :fY(y) =</p><p>|dx/dy|fX(x).</p><p>Therefore, substituting g–1(c) = (c – 10,000)/100 for h and multiplying by</p><p>images give</p><p>images</p><p>Note that care must be taken to obtain the region on the c axis in which the</p><p>density function holds. In this case it is found simply by calculating the</p><p>values of c corresponding to h = 100 and 110, the ends of the region on the h</p><p>axis. The PDF of C is also shown in Fig. 2.3.3c. In this simple linear case,</p><p>the shape of the density function has been left unchanged. As is</p><p>demonstrated in the following illustration this is not generally the case.</p><p>Illustration: Bacteria growth for random time We illustrate here</p><p>separately the two procedures, passing from CDF to CDF and passing from</p><p>PDF to PDF. Under constant environmental conditions the number of</p><p>bacteria Q in a tank is known to increase proportionately to eλT, where T is</p><p>the time, λ is the growth rate, and the proportionality constant k is the</p><p>population at time T = 0:†</p><p>images</p><p>If the time permitted for growth (as determined, say, by the time</p><p>required for the charge of water in which the bacteria are living to pass</p><p>through a filter) is a random variable with distribution function FT (t), t ≥ 0,</p><p>then the final population has distribution function</p><p>images</p><p>Here, g–1(q) = (1/λ) ln (q/k).</p><p>Now, considering passing from PDF to PDF,</p><p>images</p><p>Note that the derivative of the inverse function is a function of q, not a</p><p>constant. Then,</p><p>images</p><p>Clearly the population Q will be no smaller than the initial population</p><p>k.</p><p>Suppose, as a specific example, that fT(t) is of a decaying type, say,</p><p>images</p><p>Then</p><p>images</p><p>These two distributions are sketched in Fig. 2.3.5. Note the change in</p><p>shape in passing from one PDF to the other.</p><p>images</p><p>Fig. 2.3.5 Bacteria-growth illustration.</p><p>If the relationship between Y and X is one-to-one, but monotonically</p><p>decreasing, Y will take on values less than any particular value y0 only if X</p><p>takes on values greater than the corresponding value x0. The implications</p><p>for the previous derivations can be shown easily by the reader. The final</p><p>result for the relationship between PDF’s is simply that an absolute value</p><p>sign needs to be placed about the dx/dy or dg–1(y)/dy in previous equations,</p><p>yielding the general result:</p><p>images</p><p>images</p><p>The simplest and single most important application of this formula is for the</p><p>case when the relationship between X and Y is linear:</p><p>Simple scale changes, such as feet to inches or Fahrenheit to Centigrade, are</p><p>commonly arising examples. In this case, X = (Y – a)/b and |dx/dy| = |1/b|.</p><p>Hence,</p><p>images</p><p>The effect of such a linear transformation is to retain the basic shape of the</p><p>function, but to shift it and to expand or contract it.</p><p>General transformations Relationships Y = g(X), for which Eq. (2.3.13)</p><p>holds, are very common in engineering, where an increase in one variable</p><p>usually means either an increase or decrease in some dependent variable.</p><p>Nevertheless in many situations a more complicated nonmonotonic or non-</p><p>one-to-one relationship may be involved. The simple discrete illustration at</p><p>the beginning of this section is an example. Although the relationship was</p><p>nondecreasing, it was not one-to-one; values of X of either 0 or 1, for</p><p>example, led to a single value of Y, namely, 0. In all such problems the</p><p>reader is urged to deal directly with the cumulative distribution function of</p><p>Y, as follows.</p><p>Mathematically, the problem is to find FY(y) given that Y = g(X) and given</p><p>that X has CDF FX(x). Conceptually the problem is simple. By definition</p><p>images</p><p>but</p><p>images</p><p>where Ry is that region where g(x) is less than or equal to y. In Fig. 2.3.6a a</p><p>representation of a general function y = g(x) is sketched. Any particular</p><p>value of y, say y0, is represented by a straight horizontal line. To determine</p><p>FY(y0) is to determine the probability that the random variable X falls in any</p><p>of those intervals where the curve g(x) falls below the horizontal line, y = y0</p><p>If, for example, X were described by a continuous PDF as shown in Fig.</p><p>2.3.6b, FY(y0) would be equal to the crosshatched area under the density</p><p>function in these intervals. Solution of the problem requires relating these</p><p>intervals to each and every value y0.</p><p>Consider the following example. The kinetic energy K of a moving mass,</p><p>say a simple structural frame vibrating under a dynamically imposed wind or</p><p>vehicle load, is proportional to the square of its velocity V:</p><p>images</p><p>images</p><p>Fig. 2.3.6 Functions of random variables: general case.</p><p>in which m is the known mass of the object. Velocity may, of course, be</p><p>positive or negative (e.g., right or left, up or down). The relationship is</p><p>neither monotonic nor one-to-one, as shown in Fig. 2.3.7a. The probability</p><p>that the kinetic energy will be less than a certain value k0 is evidently</p><p>images</p><p>The probability of the complementary event K > k0 is more easily</p><p>calculated. This is 1 – FK(k):</p><p>images</p><p>Since these events are mutually exclusive,</p><p>images</p><p>This result is true no matter what the distribution of V. As a specific case,</p><p>suppose the velocity has been repeatedly measured and the engineer claims</p><p>that the triangular-shaped distribution shown in Fig. 2.3.7b represents a good</p><p>model of the random behavior of the velocity at any instant. By the</p><p>symmetry of the distribution of V,</p><p>images</p><p>The limits of the values that K can take on, namely, 0 and images follow</p><p>from the minimum (0) and maximum (3) values of the absolute magnitude</p><p>of the velocity. The density function of K can be found by differentiation of</p><p>the CDF:</p><p>images</p><p>This density function is shown in Fig. 2.3.7c.</p><p>In more complicated cases involving non-one-to-one transformations,</p><p>required regions of integration may become difficult to determine and</p><p>sketches of the problem are strongly recommended. If discrete or mixed</p><p>random variables are involved (see, for example, the pump-selection</p><p>illustration to follow in Sec. 2.4), sums may have to replace or supplement</p><p>integration, but the formulation in terms of the cumulative distribution of Y</p><p>remains unchanged. The determination of the PDF or PMF from the CDF</p><p>follows by the techniques discussed in Sec. 2.2.</p><p>2.3.2 Functions of Two Random Variables</p><p>Frequently a quantity of interest to the engineer is a function of two or more</p><p>variables which have been modeled as random variables. The implication is</p><p>that the dependent variable is also random. For example, the spatial average</p><p>velocity V in a channel is the flow rate Q divided by the area A. If Q and A</p><p>have a given joint probability distribution, the problem becomes to find the</p><p>probability law of V. The total number of vehicles Z on a particular</p><p>transportation link may be the sum of those vehicles on two feeder links, X</p><p>and Y.</p><p>images</p><p>Fig. 2.3.7 Energy-velocity illustration, Nonmonotonic Case. (a) Functional</p><p>relationship between energy and velocity; (b) given PDF of velocity V; (c)</p><p>derived PDF of kinetic energy K.</p><p>Clearly in a discrete case like the latter example we can use enumeration to</p><p>calculate the PMF or CDF of Z. The probability that Z equals any particular</p><p>value z, say 4, is the sum of the probabilities of all the pairs of values of X</p><p>and Y which sum to 4, that is, (0,4), (1,3), (2,2), (3,1), and (4,0). These</p><p>individual probabilities are given in the known joint PMF of X and Y.</p><p>Similarly the CDF of Z at value z could be found by summing the</p><p>probabilities of all pairs (x,y) for which x + y ≤ z. In difficult cases this</p><p>approach may also prove computationally advantageous when dealing in an</p><p>approximate way with continuous random variables. In this section we deal,</p><p>however, with analytical solutions of problems involving continuous random</p><p>variables.</p><p>Two approaches are considered and illustrated. One begins, as is generally</p><p>appropriate, with the determination of the CDF of the dependent variable;</p><p>the other makes use of the notions of the conditional distribution (Sec. 2.2.2)</p><p>to go directly after the PDF of the dependent random variable.</p><p>Our problem, in general, is to find FZ(z) when we know Z = g(X,Y) and</p><p>when the joint probability law of X and Y is also known.</p><p>Direct method In Sec. 2.3.1 we found the CDF of the dependent variable Z,</p><p>say, at any particular value z0 by finding the probability that the independent</p><p>variable X, say, would take on a value in those intervals of the x axis where</p><p>g(x) ≤ z0. Now, by extension, since Z is a function of both X and a second</p><p>variable Y, we must calculate FZ(z0) by finding the probability that X and Y</p><p>lie in a region where g(x,y) ≤ z0.</p><p>An easily visualized example is the following. We wish to determine the</p><p>CDF of Z, where Z is the larger of X and Y, or</p><p>images</p><p>If, for example, X and Y are the magnitudes of two successive floods or of</p><p>the queue lengths of a two-lane toll station, then primary interest may lie not</p><p>in X or Y in particular, but in the Z, the greater of the two, whichever that</p><p>might be. To determine FZ(z) for any value z, it is necessary to integrate the</p><p>joint PDF of the (assumed continuous) random variables X and Y over the</p><p>region where the maximum of x and y is less than z. This is the same as the</p><p>region where both x and y are less than z, that is, the region shown in Fig.</p><p>2.3.8.</p><p>images</p><p>images</p><p>Fig. 2.3.8 Z = max[X,Y] illustration.</p><p>The integral of fX,Y(x,y) over this region is, of course, simply FX,Y(z,z) [Eq.</p><p>(2.2.41)]. If, as a special case, X and Y are independent and identically</p><p>distributed† with common CDF FX(r),</p><p>images</p><p>In this</p><p>case it is possible to determine the PDF of Z explicitly by</p><p>differentiation</p><p>images</p><p>In words, this last result can be interpreted as the statement that “the</p><p>probability that the maximum of two variables is in a small region about z is</p><p>proportional to the probability that one variable X or Y is less than or equal</p><p>to z while the other is in a small region about z.” Since this can happen in</p><p>two mutually exclusive ways, the (equal) probabilities are added. The</p><p>treatment of the maximum of a set of several random variables will be</p><p>discussed more fully in Sec. 3.3.3.</p><p>If, for example, X and Y are independent and identically distributed annual</p><p>stream flows, with common distribution,</p><p>images</p><p>implying</p><p>images</p><p>then the distribution of Z, the larger of the two flows, is</p><p>images</p><p>and</p><p>images</p><p>These density functions are sketched in Fig. 2.3.9 for images acre-ft.</p><p>Illustration: Distribution of the quotient As a second example, consider</p><p>the determination of the probability law of Z when</p><p>images</p><p>In specific problems Z might be a cost-benefit ratio or a “safety factor,”</p><p>the ratio of capacity to load. Proceeding as before to find the CDF of Z,</p><p>assuming that X and Y are jointly distributed,</p><p>images</p><p>Fig. 2.3.9 PDF of stream flow of each of two rivers and PDF of largest</p><p>stream flow, (a) Density function of stream flow, fX(x) = 0.001e–0.001x; (b)</p><p>density function of largest stream flow.</p><p>images</p><p>where Rz is that region of the xy plane, the sample space of X and Y,</p><p>where x/y is less than z. Such a region is shown shaded in Fig. 2.3.10 for a</p><p>particular value of z. Other values of z would lead to other lines x/y = z</p><p>through the origin. The limits of integration are of the same form for any</p><p>value of z and follow from inspection of this figure:</p><p>images</p><p>images</p><p>Fig. 2.3.10 Derived-distribution illustration, Z = X/Y.</p><p>To carry out this integration may in fact be a troublesome task,</p><p>particularly as the joint PDF of X and Y may be defined by different</p><p>functional forms over different regions of the x and y plane. We shall give</p><p>complete examples later in this section, but we prefer to avoid further details</p><p>of calculus at this point as they serve only to confuse the probability</p><p>theoretic aspects of the problem, which are complete in this statement, Eq.</p><p>(2.3.31).</p><p>In this case we can again find an expression for the PDF of Z by</p><p>differentiating with respect to z before carrying out the integration. Consider</p><p>the first term</p><p>images</p><p>The second term is similar, lacking the minus sign. The terms may be</p><p>combined to form</p><p>images</p><p>Again for specific functions fX,Y(x,y) the subsequent evaluation of the</p><p>integral may be tedious, but it is not fundamentally a probability problem.</p><p>Distribution of sum X and Y by conditional distributions It is useful to</p><p>have available the important results for the case when the dependent variable</p><p>is the sum of two random variables. Sums of variables occur frequently in</p><p>practice, e.g., sums of loads, of settlements, of lengths, of waiting times, or</p><p>of travel times. Furthermore, their study is an important part of the theories</p><p>of probability and mathematical statistics, as we shall see in Chaps. 3 and 4.</p><p>In determining these results, an alternate, but less general, approach to</p><p>finding directly the formula for the PDF of a continuous function of</p><p>continuous random variables will be demonstrated. This method employs</p><p>conditional probability density functions and a line of argument that often</p><p>proves enlightening as well as productive.</p><p>We wish to determine the density function of</p><p>images</p><p>when the joint PDF of X and Y is known. Let us consider first the</p><p>conditional density function of Z given that Y equals some particular value y.</p><p>Given that Y = y, Z is</p><p>images</p><p>and the (conditional) distribution of X is</p><p>images</p><p>The density function of such a linear function of a single random variable (y</p><p>being treated as a constant for the moment) is given in Eq. (2.3.15). Using</p><p>the results found there, conditional on Y = y, the conditional PDF of Z is</p><p>images</p><p>The joint density function of any two random variables Z and Y can always</p><p>be found by multiplying the conditional PDF of Z given Y by the marginal</p><p>PDF of Y [Eq. (2.2.34)]:</p><p>images</p><p>Substituting Eq. (2.3.37),</p><p>images</p><p>But the right-hand side, being the product of a conditional and a marginal, is</p><p>only the joint PDF of X and Y evaluated at z – y and y. Thus</p><p>images</p><p>In words, this result says, roughly, that the likelihood that Y = y and Z = X +</p><p>Y = z equals the likelihood that Y = y and X = z – y.</p><p>The marginal of Z follows upon integration over all values of y [Eq.</p><p>(2.2.43)]:</p><p>images</p><p>or</p><p>images</p><p>For the important special case when X and Y are independent, their joint</p><p>PDF factors into</p><p>images</p><p>By the symmetry of the argument, it follows that it is also true that</p><p>images</p><p>or, if X and Y are independent,</p><p>images</p><p>Roughly speaking, this last equation states that the probability that Z lies in a</p><p>small interval around Z is proportional to the probability that X lies in an</p><p>interval x to x + dx times a factor proportional to the probability that Y lies in</p><p>a small interval around z – x, the value of Y necessary to make X + Y equal z.</p><p>This product is then summed over all values that X can take on. [Equation</p><p>(2.3.45) is the type of integral known as the convolution integral. This form</p><p>occurs time and time again in engineering, and it will be familiar already to</p><p>the student who has studied the dynamics of linear structural, hydraulic,</p><p>mechanical, or electrical systems.]</p><p>As mentioned above, equations such as Eqs. (2.3.24), (2.3.33), and (2.3.45)</p><p>are in fact complete answers to the probability theory part of the problem of</p><p>finding the distribution of the function of other random variables, but the</p><p>completion of the problem, that is, carrying out the indicated integrations,</p><p>may prove challenging, particularly since many practical PDF’s are defined</p><p>by different functions over different regions. Illustrations follow.</p><p>Illustration: Distribution of total waiting time When a transportation</p><p>system user must travel by two modes, say bus and subway, to reach his</p><p>destination, a portion of this trip is spent simply waiting at stops or terminals</p><p>for the arrivals of the two vehicles. Under certain conditions to be discussed</p><p>in Sec. 3.2, it is reasonable to assume that the individual waiting times X and</p><p>Y have distributions of this form</p><p>images</p><p>To determine the properties of the total time spent waiting for the</p><p>vehicles to arrive we must find the PDF of Z = X + Y. Assuming</p><p>independence of X and Y, we can apply Eq. (2.3.45):</p><p>images</p><p>Substitution of the functions takes some care, since they are, in fact,</p><p>zero for negative values of their arguments. Since fX(x) is zero for negative</p><p>values of x:</p><p>images</p><p>Since fY(y) is zero for y negative, fY(z – x) is zero for z – x negative or</p><p>for x greater than z. Therefore,</p><p>images</p><p>images</p><p>Fig. 2.3.11 Waiting-time illustration with Z = X + Y. (a) Bus waiting time,</p><p>minutes; (b) subway waiting time, minutes; (c) total waiting time, minutes.</p><p>The density functions of X, Y, and Z are sketched in Fig. 2.3.11 for α =</p><p>0.1 min–1 and β = 0.2 min–1.</p><p>Illustration: Earthquake intensity at a site We are interested as the</p><p>designers of a dam which is in an earthquake-prone region in studying the</p><p>intensity of ground motion at the site given that an earthquake occurs</p><p>somewhere in a circular region of radius r0 surrounding the site. The</p><p>information about the geology of the region is such that it is reasonable to</p><p>assume that the location of the source of the disturbance (the epicenter) is</p><p>equally likely to be anywhere in the region; that is, the engineer assigns an</p><p>equal probability to equal areas in the region and hence a constant density</p><p>function over the circle. The constant value is 1/(πr0</p><p>2), yielding a unit</p><p>volume under the function. The implication is that the density function of R,</p><p>the radial distance from the site to the epicenter,</p><p>has a triangular distribution</p><p>images</p><p>a fact the reader can verify formally by the techniques of this section.</p><p>We shall assume that historical data for the region suggest a density</p><p>function of the form†</p><p>images</p><p>for Y, the Richter magnitudes of earthquakes of significant size to be of</p><p>concern to the engineer. Magnitudes greater than 9 have not been observed.</p><p>The equation relating an empirical measure of the intensity X of the ground</p><p>motion at the site to the magnitude and distance of the earthquake we shall</p><p>assume to be</p><p>images</p><p>reflecting the attenuation of intensity with distance from the epicenter.</p><p>‡ The distribution of X is the desired result.</p><p>We seek first the CDF of X,</p><p>images</p><p>in which Rx is the region in the ry plane where g(y,r) = c1 + c2y – c3 In</p><p>r is less than x. For a given value of x this region is as shown in Fig. 2.3.12a.</p><p>For such fixed values of x, it is possible to solve for the equation of the line</p><p>bounding the region as a function of r:</p><p>images</p><p>Fig. 2.3.12 Earthquake-intensity illustration, (a) Rx: the region where g(y,r)</p><p>= c1 + c2y – c3 ln r ≤ x for a particular value of x; (b) region where fY,R(y,r)</p><p>is nonzero.</p><p>images</p><p>Hence, in general, the CDF of X is (for nonnegative random variables Y</p><p>and R) found by integrating over this region:</p><p>images</p><p>This completes the formal probability aspects of the problem. The</p><p>calculus of the integral’s evaluation proves more awkward.</p><p>In this particular case the joint density function of Y and R is positive</p><p>only over the region shown in Fig. 2.3.12b. This causes the limits of the</p><p>integrals to become far more complicated. As shown in Fig. 2.3.12 three</p><p>cases, x1, x2, or x3, exist depending on whether the particular value of x</p><p>creates a bounding curve passing out through the top of the rectangle,</p><p>through its side, or not at all. In the first situation, where x is larger than x′ =</p><p>c1 + 9c2 – c3 ln r0,</p><p>images</p><p>For x″ ≤ x ≤ x′ complicated forms of FX(x) and fX(x) result. For x ≤ x″,</p><p>both functions are zero. For the values of the constants appropriate for</p><p>Southern California (with r0 = 300 km) the CDF and PDF of X, the intensity</p><p>of the ground motion at a site given that an earthquake of magnitude 5 to 9</p><p>occurs within this radius, are plotted in Fig. 2.3.13. Notice that as long as</p><p>one is interested only in intensities in excess of 7.1 (i.e., in intensities with</p><p>probability of occurrence, given an earthquake, of less than 2.5 percent) it is</p><p>possible to deal with the distribution in the region in which its form is</p><p>simple and tractable. (Other problems will be found throughout this text</p><p>dealing with this question of seismic risk.)</p><p>images</p><p>Fig. 2.3.13 Probability distribution of intensity at site.</p><p>A variety of methods and problems in deriving distributions of</p><p>random variables will go unillustrated here. The awkard analytical aspects</p><p>far outweigh any new insights they provide into probability theory, †</p><p>2.3.3 Elementary Simulation‡</p><p>The probabilistic portion of the derivation of the distributions of a function</p><p>of one or more random variables may be a rather direct problem. The</p><p>calculus needed to evaluate the resulting integrals, however, is frequently</p><p>not tractable. In such circumstances one can often resort to numerical</p><p>integration or other techniques beyond the interest of this work. One</p><p>approximate method of solution of derived distribution problems is of</p><p>immediate interest, however, because it makes direct use of their</p><p>probabilistic nature§ to obtain artificially the results of many repeated</p><p>experiments. The histogram of these results will approximate the desired</p><p>probability distribution. This, the Monte Carlo method, or simulation, is best</p><p>presented by example.</p><p>Simulating a discrete random variable First let us find simply the</p><p>distribution of the ratio of the number of wet weeks to the number of dry</p><p>weeks in a year on a given watershed, when N, the total number of rainy</p><p>weeks with at least a trace of rain, is a random variable with a probability</p><p>mass function given by¶</p><p>images</p><p>We seek the distribution, then, of the ratio of wet to dry, or</p><p>images</p><p>We could use, following the methods of Sec. 2.3.1, simple enumeration of</p><p>all possible values of N and the corresponding values of Y to find the PMF</p><p>of Y. For illustrative purposes we are going to determine an approximation</p><p>of the distribution of Y by carrying out a series of experiments, each of</p><p>which simulates a year or an observation of N. In each experiment we shall</p><p>artificially sample the distribution of the random variable N to determine a</p><p>number n, the number of rainy weeks which occurred in that “year.” Then</p><p>we can calculate y = n/(52 – n) to find the ratio of the number of wet to dry</p><p>weeks in that “year,” that is, an observation of the random variable Y. A</p><p>sufficient number of such experiments will yield enough data to draw a</p><p>histogram (Sec. 1.1), which we can expect to approximate the shape of the</p><p>mass function pY(y). It remains to be shown how the sampling is actually</p><p>carried out.</p><p>The experimental sampling of mathematically defined random variables is</p><p>accomplished by selecting a series of random numbers, each of which is</p><p>associated with a particular value of the random variable of interest.</p><p>Random numbers are generated in such a manner that each in the set is</p><p>equally likely to be selected. The numbers 1 to 6 on a perfect die are, for</p><p>example, a set of random numbers. Divide the circumference of a dial into</p><p>10 equal sectors labeled 0 to 9 (or 100 equal sectors labeled 0 to 99, etc.),</p><p>attach a well-balanced spinner, and you have constructed a mechanical</p><p>random number generator. Various ingenious electrical and mechanical</p><p>devices to generate random numbers have been developed, and several</p><p>tables of such numbers are available (see Table A.8), but most random</p><p>number generation in practice is accomplished on computers through</p><p>numerical schemes, †</p><p>To relate the value of the random variable of interest N to the value obtained</p><p>from the random number generator, one must assign a table of relationships</p><p>(a mapping) which matches probabilities of occurrence. Assume that a</p><p>random number generator is available which produces any of the numbers 0</p><p>to 9999 with equal probability 1/10,000. On the other hand, the probability</p><p>that N takes any of the various values 0, 1, 2, . . ., 52 can be determined by</p><p>evaluating its probability mass function. For example,</p><p>Then values of the random numbers and the random variable N might be</p><p>assigned as follows:</p><p>If the first random number generated happened to be 4751, the number n of</p><p>rainy weeks in that simulated year would be taken as 20. Then the</p><p>corresponding value of Y would be Repetition of these</p><p>“experiments,” a process ideally suited for digital computers, will yield y1,</p><p>y2, y3,, a sample of observed values of Y. A histogram (Sec. 1.1) of these</p><p>values will approximate the true PMF of Y. For example, the 100 random</p><p>numbers in Table 2.3.1 lead to the indicated values of N, Y, and the</p><p>histogram in Fig. 2.3.14. The exact PMF is plotted in the same figure.</p><p>Fig. 2.3.14 Probability mass functions of Y, exact and by simulation.</p><p>Note that if another set of 100 values were generated, a histogram similar in</p><p>shape but different in details is to be expected. In short, the histograms are</p><p>themselves observations of random phenomena. Practical questions which</p><p>the simulator must face are, “How much variability is there in such</p><p>histograms?” “What can I safely conclude about the true PMF from one</p><p>observed histogram?” “How does the shape of an observed histogram</p><p>behave probabilistically as the sample size increases?” These and similar</p><p>questions will be considered in Secs. 4.4 and 4.5.</p><p>Illustration: Total annual rainfall Monte Carlo techniques become</p><p>particularly useful when the relationships among variables are complicated</p><p>functions, involve many random variables, or, as in the next example,</p><p>themselves depend upon a random element. In</p><p>this illustration it is desired to</p><p>determine the distribution of total annual rainfall T when the rainfall Ri in</p><p>the ith rainy week is given by</p><p>and when the total number of rainy weeks in a year is N, as before, a</p><p>random variable with the distribution given in Eq. (2.3.52). This problem</p><p>might be part of a large study of the performance of a proposed irrigation</p><p>system. The relationship between T, N, and the Ri’s is:</p><p>That is, T is the sum of a random number of random variables. (It is</p><p>assumed, for convenience, that rainfalls in rainy weeks are identically</p><p>distributed, mutually independent, and independent, too, of the number of</p><p>rainy weeks in the year.)</p><p>In Sec. 2.3.2, we saw that the determination of the sum of even two</p><p>random variables can be difficult enough. Analytical solutions to the type of</p><p>problem here, are, in fact, feasible (Sec. 3.5.3), but we propose to determine</p><p>the distribution of T experimentally by simulating a number of years and</p><p>producing a sample of values of T. In each year we shall sample the random</p><p>variable N to determine a particular number of rainy weeks n. Then the</p><p>random variables Ri will be sampled that number of times and the n values</p><p>summed to get one observation of T. This total experiment will be repeated</p><p>as accuracy requires.</p><p>The sampling of the discrete variable N has been discussed. Several</p><p>schemes for relating values of random numbers to values of the continuous</p><p>random variable Ri can be considered. One approach simply approximates</p><p>the continuous distribution of Ri by a discrete one. Equal intervals of r, say</p><p>0.0 to 0.9, 0.10 to 0.19, etc., might be considered, the probability of Ri</p><p>taking on a value in each interval evaluated, and a corresponding proportion</p><p>of the random numbers assigned to be associated with the interval. The</p><p>middle value† of each interval can then be used as the single representative</p><p>value of the entire interval. If a random number associated with the interval</p><p>were drawn, this representative value would be accumulated with others to</p><p>determine an observed value of T.</p><p>Table 2.3.1</p><p>Alternatively, a preferable scheme calls for dividing the range of r</p><p>into intervals of equal probability. The fineness of the intervals chosen</p><p>depends only on the accuracy desired.† Assume here that 10 intervals are</p><p>considered sufficient. Then the dividing lines between intervals are 0, r1, r2,</p><p>. . ., r10 such that</p><p>From tables of e–x we obtain intervals:</p><p>Corresponding values of the random numbers are assigned as shown.</p><p>In this case, where the number of intervals is small, the</p><p>determination of the representative value for each interval is most</p><p>important. What, for example, should be used as the representative value of</p><p>the last interval? The center of gravity is a logical choice. The centroid of</p><p>the segment of the density function over the last interval is</p><p>images</p><p>images</p><p>Fig. 2.3.15 Partition of the distribution of rainfall R for simulation. (a) CDF</p><p>of R; (b) PDF of R.</p><p>The use here of these centroids will be justified in the following</p><p>section, where it will be known as the “conditional mean” of Ri given that</p><p>1.15 ≤ Ri ≤ ∞. This and other representative values appear in the previous</p><p>list, and the dissection of the PDF and CDF of R is illustrated in Fig. 2.3.15a</p><p>and b.</p><p>In computer-aided simulation a large number of intervals are</p><p>normally used (e.g., 1 million, if six-digit random numbers are generated).</p><p>In this case it is common to use simply the division values rj as the</p><p>representative values images A random number is generated on the</p><p>interval 0 to 1, say 0.540012. The corresponding value r is found by solving</p><p>F(r) = 0.540012 for r, that is, by finding the inverse of F(r). In this example</p><p>we have</p><p>images</p><p>implying</p><p>images</p><p>This same procedure is valid no matter what the form of the CDF of the</p><p>random variable, † It is not always feasible, however, to obtain a closed-</p><p>form analytical expression for the inverse of the CDF of a random variable.</p><p>In this case one can find the value of x corresponding to the randomly</p><p>selected value of FX(x) by hand through graphical means or by computer</p><p>through “table lookup” schemes. Various other techniques are also used to</p><p>generate sample values of random variables with particular distributions. For</p><p>example, a sample of a “normally distributed” random variable (Sec. 3.3.1)</p><p>is usually obtained in computer simulation by adding 12 or more random</p><p>numbers. The justification for this procedure will become clear in Sec. 3.3.1.</p><p>In the first experiment in this example a random number 4751 was</p><p>generated, and a sample value of N, n = 20, determined. Consequently, to</p><p>complete this first experiment, 20 random numbers must be generated and</p><p>the corresponding values of the Ri listed and accumulated to determine a</p><p>sample value of T. Such a sequence is shown in Table 2.3.2 using the</p><p>divisions shown in Fig. 2.3.15. The table represents “1 year” of real-time</p><p>observation.</p><p>Again, repetition of such experiments will yield a sample of values</p><p>of T whose histogram is an approximation of fT(t) and whose frequency</p><p>polygon † is an approximation to FT(t). Figure 2.3.16 shows the results of</p><p>such a series of experiments for several sample sizes and two different</p><p>numbers of histogram intervals of r. Analytical treatment of aspects of this</p><p>same problem will follow in Sec. 2.4 and Chap. 3.</p><p>Table 2.3.2 Simulated values of the rainfall in the 20 rainy weeks of 1</p><p>year</p><p>images</p><p>images</p><p>Fig. 2.3.16 Histograms of several simulation runs reduced to estimated</p><p>density functions and an estimated cumulative distribution function.</p><p>It should be evident even from these simple illustrations that many complex</p><p>engineering problems can be analyzed by Monte Carlo simulation</p><p>techniques. The models need not be restricted to functional relationships, but</p><p>can involve complicated situations in which the distributions of variables</p><p>depend upon the (random) state of the physical system or in which the whole</p><p>sequence of steps to be followed may depend upon which particular value of</p><p>a random variable is observed. Elaborate probabilistic models of many</p><p>vehicles on complicated networks of streets and intersections have proved</p><p>most useful to traffic engineers trying to predict the performance of a</p><p>proposed design. The uncertainty and variation in drivel reaction time,</p><p>driving habits, and origin-destination demands can all be treated on a</p><p>probabilistic basis. Long histories of the operation of whole river-basin</p><p>systems of water-resource controls have been simulated in minutes</p><p>(Hufschmidt and Fiering [1966]). Rainfall and runoff values form</p><p>probabilistic inputs to chains of dams and channels (real or designed) whose</p><p>proposed operating policies (e.g., the degree to which reservoirs should be</p><p>lowered in anticipation of flood runoffs) are being evaluated for long-term</p><p>consequences, likelihoods of poor performance, and economics. Simulation,</p><p>combined with numerical integration of the equations of motion, has been</p><p>used to obtain approximate distributions of the maximum dynamic response</p><p>of complex, nonlinear structures to the chaotic, random ground motions</p><p>during earthquakes (Goldberg et al. [1964]).</p><p>The successful application of simulation depends upon the appropriateness</p><p>of the model and the interpretation of the results as much as on the</p><p>sophistication of the simulation techniques used. The former problems are</p><p>the ones the engineer usually faces in using mathematical models of natural</p><p>phenomena; for the latter problems, those of technique, the engineer can find</p><p>help and documented experience in a number of references. (See, for</p><p>example, Tocher [1963] or Hammersley and Hands-comb [1964].) Many</p><p>methods are available, for example, to generate random numbers, to account</p><p>for dependence among variables, or to reduce the effort needed to get the</p><p>desired accuracy.</p><p>2.3.4 Summary</p><p>This section presents methods for deriving the distributions of random</p><p>variables which are functionally dependent upon random variables whose</p><p>distributions are known. In general, one should seek the CDF of the</p><p>dependent random variable Z. For any particular value z, FZ(z) is found by</p><p>calculating in the sample space of X (or X and Y) the probability of all those</p><p>events where g(x) [or g(x,y)] is less than or equal to z:</p><p>images</p><p>This can be done by enumeration if the distribution of X (or X and Y) is</p><p>discrete. If the distribution of X is continuous, integration is required. In</p><p>certain circumstances (monotonic, increasing, one-to-one relationships), this</p><p>procedure simply reduces to</p><p>images</p><p>The density function of Z can be found by differentiation of FZ(z)</p><p>(assuming that Z is a continuous random variable). In certain cases the</p><p>differentiation can be carried out explicitly before particular probability laws</p><p>FX(x) are considered, in which case one can obtain a formula for the PDF of</p><p>Z directly. The two most important examples are</p><p>1. Under monotonic, one-to-one conditions:</p><p>images</p><p>2. If Z = X + Y :</p><p>images</p><p>Certain computational methods are available for cases where analytical</p><p>solutions are difficult or impossible. They include</p><p>1. Approximating continuous distributions by discrete ones, and solving by</p><p>enumeration</p><p>2. Applying simulation techniques to obtain, by repeated experimentation,</p><p>observed histograms which approximate desired results.</p><p>2.4 MOMENTS AND EXPECTATION</p><p>Because of the very nature of a random variable it is not possible to predict</p><p>the exact value that it will assume in any particular experiment, but a</p><p>complete description of its behavior is contained in its probability law, as</p><p>presented in the CDF (or PMF or PDF, if applicable) of the variable. This</p><p>complete information can be communicated only by stipulating an entire</p><p>function, e.g., the PDF. In many situations this much information may not be</p><p>necessary or available. More concise descriptors, summarizing only the</p><p>dominant features of the behavior of a random variable, are often sufficient</p><p>for the engineering purpose at hand. One or more simple numbers are used</p><p>in place of a whole probability density function. These numbers usually take</p><p>the form of weighted averages of certain functions of the random variable.</p><p>The weights used are the PMF or PDF of the variable, and the average is</p><p>called the expectation of the function.</p><p>We will find that, compared with entire probability laws, these expectations</p><p>are much easier to work with in the analysis of uncertainty, as well as much</p><p>easier to obtain estimates of from available data. Therefore, in engineering</p><p>applications, where expedience often dictates that approximate but fast</p><p>answers are better than none at all, averages and expectations prove</p><p>invaluable.</p><p>2.4.1 Moments of a Random Variable</p><p>Mean Every engineer is familiar with averages of observed numerical data.</p><p>The sample mean and sample variance (Sec. 1.2) are the most common</p><p>examples. Although they do not communicate all the available information,</p><p>they are concise descriptors of the two most significant properties of the</p><p>batch of observed data, namely, its central value and its scatter or dispersion.</p><p>On the other hand, given a solid body, e.g., a rod of nonuniform shape or</p><p>density, the engineer is accustomed to determining certain numerical</p><p>descriptions of the body such as the location of its center of mass and a</p><p>moment of inertia about that point. Not complete in their description, these</p><p>quantities are nonetheless sufficient to enable the engineer to predict a great</p><p>deal about the gross static and dynamic behavior of the body.</p><p>Both these examples deserve being kept in mind when we define the mean</p><p>mX or the expected value E[X] of a discrete random variable X as</p><p>images</p><p>or, for a continuous random variable, as</p><p>images</p><p>In the mean (or mean value) we are condensing the information in the</p><p>probability distribution function into a single number by summing over all</p><p>possible values of X the product of the value x and its likelihood pX(x) or</p><p>fX(x) dx.</p><p>Recall from Chap. 1 that the sample mean of n numbers was defined as</p><p>images</p><p>If several observations of each value xi are found, this definition can be</p><p>written</p><p>images</p><p>in which r is the total number of distinct values observed, ni is the total</p><p>number of observations at value xi, and fi = ni/n is the observed frequency of</p><p>the value xi in the sample.</p><p>Notice the close proximity in appearance between the definition of the mean</p><p>of a (discrete) random variable, Eq. (2.4.1), and the sample mean of a batch</p><p>of observed numbers, Eq. (2.4.4). This similarity helps make clear the notion</p><p>of the mean of a random variable (especially when repeated observations of</p><p>the random variable are anticipated), but the student should be most careful</p><p>to avoid confusing the two means. The sample mean is computed from given</p><p>observations, and the mean (or expectation) is computed from the</p><p>mathematical probability law (e.g., from the PDF, CDF, or PMF) of a</p><p>random variable. The latter is sometimes called the population mean to</p><p>distinguish it from the sample mean. Geometrically, it is clear from its</p><p>definition, Eqs. (2.4.1) or (2.4.2), that the mean defines the center of gravity</p><p>of the shape defined by the PDF or PMF of a random variable.</p><p>In a physical problem, where some phenomenon has been modeled as a</p><p>random variable, the mean value of that variable is usually the most</p><p>significant single number the engineer can obtain. It is a measure of the</p><p>central tendency of the variable, and, often, of the value about which scatter</p><p>can be expected if repeated observations of the phenomenon are to be made.</p><p>The sample mean of many such observations will, with high probability, †</p><p>be very close to the (population) mean of the underlying random variable.</p><p>For these and other reasons to be seen, when probabilistic model and a</p><p>deterministic (nonprobabilistic) model of a physical phenomenon are</p><p>compared, it is usually the mean of the probabilistic model which one</p><p>compares with the single value of a deterministic model.</p><p>The mean of the discrete random variable adopted in Sec. 2.2.1 to model the</p><p>annual runoff (see Fig. 2.2.3) is computed as follows, using Eq. (2.4.1):</p><p>images</p><p>The mean runoff is 4167 acre-ft. This is a central value, a value about which</p><p>observed values of X will tend in the long run to be scattered. It is probably</p><p>the number the engineer would use if he were restricted to using only a</p><p>single number to describe the runoff, that is, if he had to treat runoff</p><p>deterministically rather than probabilistically in his analysis.</p><p>The mean value of the rainfall in a rainy week, the random variable R in the</p><p>illustration in Sec. 2.3.3, is calculated by integration, after Eq. (2.4.2).</p><p>Substituting for the PDF the function 2e–2r, r ≥ 0,</p><p>images</p><p>images</p><p>If it rains in a week, ½ in. is the mean value or “expected” value of the</p><p>rainfall. Comparing this value with the density function of R (Fig. 2.3.15), it</p><p>is apparent that in this case, the mean of R is not central, in that it does not</p><p>correspond to a peak in the distribution. Nor is the mean the value which</p><p>will be exceeded half of the time.† (That is, 1 – FR(½ in.) = e–1 = 0.368 ≠</p><p>0.5. Solving 1 – FR(u) = 0.5 yields the median u = 0.346 in.) Nonetheless,</p><p>even here the mean value yields “order of magnitude” information as to</p><p>weekly rainfall in rainy weeks, and, if the observed records of many such</p><p>weeks are averaged, this sample average would almost certainly be very</p><p>close ‡ to ½ inch (assuming always that the mathematical model is a good</p><p>representation of the physical phenomenon).</p><p>Variance The mean describes the central tendency of a random variable, but</p><p>it says nothing of that behavior which leads engineers to study probability</p><p>theory at all, namely, uncertainty or randomness. Here we seek a descriptor,</p><p>a single number, which will give an indication of the scatter or dispersion,</p><p>or, loosely, of the “randomness” in the random variable’s behavior.</p><p>Several such measures</p><p>are possible. The range of the random variable is one</p><p>example, although it is frequently a rather uninformative pair of numbers</p><p>such as – ∞ to ∞ or 0 to ∞. Even if the range is two finite numbers, say, a to</p><p>b, it gives no indication of the relative frequency of extreme values as</p><p>compared with central values. It is desirable therefore to measure the</p><p>dispersion from a central value, the mean, and to weight all deviations from</p><p>the mean by their relative likelihoods.</p><p>The most common and most useful such measure of the dispersion of a</p><p>random variable is the variance σX</p><p>2, or Var [X], It is defined as the weighted</p><p>average of the squared deviations from the mean:</p><p>images</p><p>or</p><p>images</p><p>The variance of a random variable bears the same relationship to the sample</p><p>variance of a set of numbers (Sec. 1.2) as the mean does to the sample mean,</p><p>and a comparison analogous to that made above between Eqs. (2.4.1) and</p><p>(2.4.4) could be made with ease. A more meaningful analogy to draw is that</p><p>between the variance of a random variable and the central moment of inertia</p><p>of a bar of variable density (and unit mass). The variance σX</p><p>2 is the second</p><p>central moment of the area of the PDF or PMF with respect to its center of</p><p>gravity mX.</p><p>In Fig. 2.4.1 are shown probability density functions of the same basic</p><p>shape. The curves in Figs. 2.4.1a and b differ only in their mean, while the</p><p>curves in Figs. 2.4.1b and c differ only in their variances. Smaller variances</p><p>generally imply “tighter” distributions, less widely spread about the mean.</p><p>Standard deviation The positive square root of the variance is given the</p><p>name standard deviation:</p><p>images</p><p>The conventional form of the standard notation, σ and σ 2, seems to indicate</p><p>that in practice the standard deviation is given more importance than the</p><p>variance. That is, in fact, the case.† The standard deviation has the same</p><p>units as the variable X itself and can be compared easily and quickly with</p><p>the mean of the variable to gain some feeling for the degree and gravity of</p><p>the uncertainty associated with the random variable.</p><p>images</p><p>Fig. 2.4.1 Changes in PDF’s with changes in means and standard deviations.</p><p>Coefficient of variation A unitless characteristic that formalizes this</p><p>comparison and that also facilitates comparisons among a number of random</p><p>variables of different units is the coefficient of variation VX:</p><p>images</p><p>The coefficient of variation of the strength of the concrete produced by a</p><p>given contractor is often assumed to be a constant for all mean strengths and</p><p>to be a measure of the quality control practiced in his work (see ACI [1965]</p><p>and Prob. 2.45). Thus a contractor practicing “good” control might produce</p><p>concrete with a coefficient of variation of 0.10 or 10 percent, implying that</p><p>concrete of mean strength 4000 psi would have a standard deviation of 400</p><p>psi, whereas “5000-psi concrete” would have a standard deviation of 500</p><p>psi.</p><p>The variance of the discrete runoff random variable can be computed from</p><p>Eq. (2.4.7) as follows:</p><p>images</p><p>The standard deviation of this variable is</p><p>images</p><p>and the coefficient of variation is</p><p>images</p><p>The variance of the rainfall variable R is found by applying Eq. (2.4.8):</p><p>images</p><p>while</p><p>images</p><p>and</p><p>images</p><p>Illustration: Sigma bounds and the Chebyshev inequality It is common</p><p>in engineering applications of probability theory to speak of the “one-, two-,</p><p>and three-sigma” bounds of a random variable. The range between the two-</p><p>sigma bounds, for example, is the range between mX – 2σX and mX + 2σX.</p><p>The two-sigma bounds on the runoff variable X are 4167 – 2(2200) and 4167</p><p>+ 2(2200) or 0 and 8567.† The three sets of bounds for this variable are</p><p>shown in Fig. 2.4.2.</p><p>images</p><p>Fig. 2.4.2 Sigma bounds: runoff model.</p><p>In absence of the knowledge of the complete PDF, but knowing</p><p>mean and variance, it is frequently stated in engineering applications (for</p><p>reasons that will be clear in Sec. 3.3.1) that the probability that a variable</p><p>lies within the one-sigma bounds of its mean is approximately 65 percent;</p><p>within the two-sigma bounds, about 95 percent; and within the three-sigma</p><p>bounds, about 99.5 percent. That even rough probability figures can be</p><p>given with so little information is indicative of the value of the standard</p><p>deviation and the variance as measures of dispersion. In fact, these</p><p>approximate statements should only be used when it is known that the</p><p>distribution is roughly bell-shaped.</p><p>More formally we can show that the mean and standard deviation</p><p>alone are sufficient to make certain exact statements on the probability of a</p><p>random variable lying within given bounds. The Chebyshev inequality ‡</p><p>states that</p><p>images</p><p>Note that, corresponding to the one-, two-, and three-sigma bounds, h</p><p>equals 1, 2, and 3.</p><p>For example, the runoff variable has mean 4167 and standard</p><p>deviation 2200. The Chebyshev inequality states</p><p>images</p><p>If h = 2</p><p>images</p><p>Or, recognizing in this particular case that X, the runoff, is nonnegative,</p><p>images</p><p>or, in another form,</p><p>images</p><p>The rule of thumb for the two-sigma bounds suggests that the former</p><p>probability, that is, P[X ≤ 8567], should be about 95 percent, while the exact</p><p>value, Fig. 2.4.2, is images.</p><p>The Chebyshev inequality does not yield very sharp bounds.</p><p>[Consider, for example, any value of h less than 1, when the right-hand side</p><p>of Eq. (2.4.11) is negative.] It has the advantage, however, that it requires no</p><p>assumption on the part of the engineer regarding the shape of the</p><p>distribution. Hence the probability statements are totally conservative,</p><p>within the qualification, of course, that mX and σX are known with certainty.</p><p>When these parameters’ estimates are based on only a small amount of</p><p>observed data, they, in fact, are not known with high confidence; this</p><p>uncertainty is a subject of Chaps. 4 and 6.</p><p>A host of less general, but more precise inequalities are available.</p><p>(See, for example, Parzen [1960], Freeman [1963].) As the engineer makes</p><p>more and more assumptions regarding the shape of the distribution, such as</p><p>its being unimodal (single-peaked), having “high-order contact” with the x</p><p>axis in the extreme tails, † being symmetrical, having known values of</p><p>higher moments (see Sec. 2.4.2), etc., the sharper his probability statements</p><p>can be. Finally, of course, if he is willing to stipulate FX(x) itself, he can</p><p>state exactly the proportion of the probability mass lying inside or outside</p><p>any interval. This progression is common in applied probability theory; the</p><p>more the engineer is willing to assume to be known, given, or hypothesized</p><p>information, the more precise and penetrating can he be in his subsequent</p><p>probabilistic analysis.</p><p>Discussion It should be emphasized that from the point of view of</p><p>applications, two quite opposite problems have been discussed in this</p><p>section. The first is that of calculating the mean and variance of a random</p><p>variable knowing its distribution, and the second is that of making some</p><p>kind of statement about the behavior of the variable when only the mean and</p><p>variance are known.</p><p>The latter case is frequent. It arises in many situations. Data is often</p><p>available only as summarized by its sample mean and sample variance,</p><p>which are the natural estimates of the corresponding two model parameters.</p><p>Also, in the analysis of complex models, the mean and variance of the</p><p>dependent random variable are often easily obtainable when the complete</p><p>story, i.e., the whole distribution, is lost in a maze of intractable integrals</p><p>(Sec. 2.3). As will be discussed later in this section, we can often determine</p><p>these two parameters as simple functions of the same two parameters of the</p><p>independent random variables involved, implying that the engineer may</p><p>never need to commit himself to unnecessarily detailed models of these</p><p>variables. As shown in the next section, the mean and variance alone contain</p><p>a substantial amount of information on which to base engineering decisions.</p><p>Finally, they represent the first and most important step beyond deterministic</p><p>engineering models in that they characterize not only the typical value, as</p><p>the traditional model does, but also the dispersion. Recognition of the</p><p>existence of the variance indicates that the probabilistic aspects of the</p><p>engineering problem won’t be ignored. Study of the mean and variance, and</p><p>their later counterparts, is referred to as a second-order-moment analysis;</p><p>there will be an emphasis on this notion throughout our study of applied</p><p>probability and statistics.</p><p>Summary In Sec. 2.4.1 we have defined the mean, variance, standard</p><p>deviation, and coefficient of variation of a random variable. They are</p><p>defined as sums of the possible values of the random variable or as sums of</p><p>the squared deviations from the mean, weighted by their probabilities of</p><p>occurrence. The mean and variance are the center of gravity of the</p><p>probability mass and its moment of inertia. They can be calculated from</p><p>given probability distributions, or used alone, without knowledge of the</p><p>entire distribution, as summaries of the predominant characteristics of</p><p>central value and dispersion.</p><p>2.4.2 Expectation of a Function of a Random Variable</p><p>In Sec. 2.3 we emphasized the importance in engineering applications of</p><p>being able to determine the behavior of a (dependent) random variable Y,</p><p>which is a function g(X) of another (independent) random variable X, whose</p><p>behavior is known. The computational difficulties involved in actually</p><p>finding, say, the PDF of Y from the PDF of X have also been encountered.</p><p>Fortunately, as mentioned above in the justification for a concentrated study</p><p>of the mean and variance, no such computational complications arise if one</p><p>seeks only these two moments of Y, given the probability law of the</p><p>independent variable X. Often even less is needed; for example, often only</p><p>the mean and variance of X can be used to find corresponding moments of Y.</p><p>Expectation of a function If we know the probability law of X and our</p><p>interest is in Y, where</p><p>images</p><p>then the expected value or expectation of Y is, by definition [Eq. (2.4.2)],</p><p>images</p><p>The method for computing fY (y) and its attendant difficulties were the</p><p>subject of Sec. 2.3. It is a fundamental, but difficult to prove, † result of</p><p>probability theory that this expectation can be evaluated by the much easier</p><p>computation‡</p><p>images</p><p>where the notation E[g(X)] is defined as:</p><p>images</p><p>For a discrete variable, § the expectation of g(X) is defined as</p><p>images</p><p>It is helpful to see a simple discrete illustration. Suppose a random variable</p><p>X has PMF as shown in Fig. 2.4.3a. Then the PMF of Y = g(X) = X2 is as</p><p>shown in Fig. 2.4.3b. This result can be verified by straightforward</p><p>enumeration in this simple example. The expected value of Y is, by</p><p>definition, E[Y] = (⅔)(1) + (⅓)(4) = 2. Equation (2.4.14) states, however,</p><p>that this expected value can also be found without first determining PY(y)</p><p>Using Eq. (2.4.14),</p><p>images</p><p>as before.</p><p>Notice that the two quantities defined in Sec. 2.4.1, the mean and variance,</p><p>can in fact be interpreted as merely special cases of Eq. (2.4.14) with g(X) =</p><p>X for the mean and g(X) = (X – mX)2 for the variance. This equivalence</p><p>suggests that two quite different interpretations might be given to Eq.</p><p>(2.4.14). In some situations one may think of E[g(X)] as representing the</p><p>mean of a random variable Y = g(X) conveniently calculated by this</p><p>equation rather than by Eq. (2.4.12). This interpretation is common when</p><p>interest centers on Y = g(X) as a dependent random variable functionally</p><p>related to another random variable X (with known distribution function), as,</p><p>for example, when X is velocity and Y = aX2 is kinetic energy. On the other</p><p>hand, if interest centers on X itself, then the expectation of g(X) as given by</p><p>Eq. (2.4.14) is usually interpreted as weighted average of g(X), over X, that</p><p>is, as the sum of the values of the function of X evaluated at the various</p><p>possible values of X and weighted by the likelihoods of those values of X.</p><p>The variance of X, for example, is usually thought of in this light, rather than</p><p>as the mean of a random variable Y = (X – mX)2. The distinction between</p><p>these two interpretations is not of fundamental importance, but it may prove</p><p>helpful when considering the uses to which the material to follow will be</p><p>put.</p><p>images</p><p>Fig. 2.4.3 PMF’s of X and derived Y = X2 (a) Discrete PMF of X; (b)</p><p>discrete PMF of Y = X2.</p><p>Moments In keeping with the latter interpretation of E[g(X)] as a weighting</p><p>of g(x) by fX(x), we introduce a family of averages of X, called moments,</p><p>that prove useful as numerical descriptors of the behavior of X. We call</p><p>images</p><p>the nth moment of X. Notice that this moment corresponds to the nth</p><p>moment of the area of the PDF with respect to the origin. If n = 1, when the</p><p>superscript is usually omitted, we have the mean of the variable.</p><p>It is possible, of course, to consider moments of areas about any point. In</p><p>particular, moments with respect to the mean are called central moments:</p><p>images</p><p>Such moments correspond to the familiar moments of areas with respect to</p><p>their centroids. The most important particular case is for n = 2, when</p><p>images the variance. The first central moment is, of course, always zero. If</p><p>the asymmetry of a distribution is of interest, this property is often</p><p>quantified or characterized by the third central moment μX</p><p>(3), or by the</p><p>corresponding dimensionless coefficient of skewness γ1:</p><p>images</p><p>If a distribution is symmetrical, this coefficient is zero (although the</p><p>converse is not necessarily true). Positive values of γ1 usually correspond to</p><p>PDF’s with dominant tails on the right; negative values to long tails on the</p><p>left (see Fig. 2.4.4).</p><p>A less common coefficient γ2, the coefficient of kurtosis (flatness), is</p><p>defined similarly:</p><p>images</p><p>It is often compared to a “standard value” of 3.†</p><p>images</p><p>Fig. 2.4.4 Variation of shape of PDF with coefficient of skewness γ1. (a)</p><p>Negative skewness; (b) zero skewness; (c) positive skewness.</p><p>For example, the central moments of the rainfall variable R, defined in Sec.</p><p>2.3.3, are, in general,</p><p>images</p><p>In the previous section, it was found that</p><p>images</p><p>Also</p><p>images</p><p>and</p><p>images</p><p>Hence the skewness and kurtosis coefficients are</p><p>images</p><p>indicating a positive skewness or long right-hand tail (see Fig. 2.3.15), and</p><p>images</p><p>Properties of expectation No matter which interpretation of the expectation</p><p>E[g(X)] is involved, several general properties of the operation can be</p><p>pointed out. For example, the expectation of a constant c is just the constant</p><p>itself. This fact is easily shown. Simply by writing out the definition, we</p><p>have</p><p>images</p><p>Similarly the following properties can be verified with ease for constants a,</p><p>b, and c:</p><p>images</p><p>images</p><p>images</p><p>The implication of this last equation is that expectation, like differentiation</p><p>or integration, is a linear operation. This linearity property is very useful</p><p>computationally. It can be used, for example, to find the following formula</p><p>for the variance of a random variable in terms of more easily calculated</p><p>quantities.</p><p>images</p><p>Expanding the square,</p><p>images</p><p>Each term in the sum can be treated separately as a function of X and,</p><p>according to Eq. (2.4.23), the expectation of their sum is the sum of their</p><p>expectations:</p><p>images</p><p>Using other properties of expectation [Eqs. (2.4.20) and (2.4.21)],</p><p>images</p><p>But E[X] = mX; therefore</p><p>images</p><p>In an alternate form, the variance is said to be the “mean square” minus the</p><p>“squared mean”:</p><p>images</p><p>Given the PDF of X, the simplest way to evaluate the variance is usually to</p><p>calculate images that is, E[X2], and then to subtract the squared mean.</p><p>Note that this last derivation took place without any reference to a particular</p><p>form of the PDF and even without indication as to whether a discrete or</p><p>continuous variable was involved. It is this ability to work with expectation</p><p>without specifying the PDF that often permits us to determine relationships</p><p>among the moments of two functionally related</p><p>variables, X and Y = g(X),</p><p>before specifying or even without knowledge of the PDF of X. For example,</p><p>in concrete the quadratic relationship</p><p>images</p><p>holds between compressive stress Y and unit strain X well beyond the linear</p><p>elastic range (Hognestad [1951]). If the unit strain applied to a specimen by</p><p>a testing machine is a random variable, owing, say, to uncertainties in</p><p>recording, then so is the stress in the concrete. How might the expected or</p><p>mean stress be determined? If the PDF of X is known, the mean stress mY</p><p>could be found by using the methods of Sec. 2.3 to find fY(y), and then</p><p>applying Eq. (2.4.2). Alternatively, E[Y] could be calculated using Eq.</p><p>2.4.14, since</p><p>images</p><p>But, if the mean and variance of the strain are available, even if the PDF is</p><p>not, the mean of Y can be found much more directly as simply</p><p>images</p><p>Using Eq. (2.4.24),</p><p>images</p><p>Therefore</p><p>images</p><p>Thus the mean and variance of X are sufficient to determine the mean of Y.</p><p>An important practical distinction between deterministic and probabilistic</p><p>analysis is also illustrated by this example. In a deterministic formulation of</p><p>this problem, one would assume only a single value of strain was possible,</p><p>say the typical value mX. The predicted stress would then be bmX – cmX</p><p>2,</p><p>which does not coincide with the mean value of F, unless, of course, the</p><p>dispersion in X is truly zero. The greater the uncertainty in the strain, the</p><p>greater the systematic error in the predicted value of the stress.</p><p>Formalizing this observation: in general, we cannot find the expectation of a</p><p>function of X by substituting in the function E[X] for X, or</p><p>images</p><p>For example, the mean of 1/X is not 1/mX.</p><p>The linearity property of expectation, which made the previous derivations</p><p>so direct, does not carry over to variances. The variance of Y = 2X is not two</p><p>times the variance of X. This fact is easily demonstrated:</p><p>images</p><p>Several useful general properties of variances can be stated (and easily</p><p>verified) however:</p><p>images</p><p>That is, a constant has no variance, and the standard deviation of a linear</p><p>function of X, a + bX, is just |b|σX A simple example of such a linear</p><p>function is a problem in which there is a change of units.</p><p>Conditional expectation In Sec. 2.2 we discussed briefly the construction</p><p>of conditional distributions which are formed conditionally on the</p><p>occurrence of some event. For example, in a design situation we might be</p><p>interested in the distribution of the demand or load X, given that it is larger</p><p>than some threshold value x0, say, the nominal design demand. Then, by</p><p>definition of conditional probabilities,</p><p>images</p><p>The conditional PDF is found by differentiation with respect to x:</p><p>images</p><p>The conditional distributions satisfy all the necessary conditions to be proper</p><p>probability distributions, and may be used as such.</p><p>It is therefore also meaningful to consider conditional means, conditional</p><p>variances, and in general any expectation that is conditional on the</p><p>prescribed event. So the conditional mean of the demand X, given that it is</p><p>larger than the nominal demand x0, is</p><p>images</p><p>and its conditional variance is defined as</p><p>images</p><p>In general, for some event A,</p><p>images</p><p>Note that there is a trivial case:</p><p>images</p><p>A numerical example of the use of conditional distributions and expectations</p><p>in engineering design will be found in an illustration to follow.</p><p>Expected costs and benefits A common use of expectation of a function of</p><p>a random variable arises from the practice of basing engineering decisions,</p><p>in situations involving risk, on expected costs. Frequently a portion of the</p><p>total cost of a proposed design depends upon the more or less uncertain</p><p>future magnitudes of such phenomena as local rainfall, traffic volume, or</p><p>unit bid prices. If the engineer describes the uncertainty associated with</p><p>these variables by treating them as random variables, then, when comparing</p><p>alternate designs, the question arises of how to combine those portions of the</p><p>costs or benefits that depend upon these random variables with the other,</p><p>nonrandom, components of cost. How do you compare a more expensive</p><p>spillway design with one of smaller initial cost, but of smaller capacity, and</p><p>hence of greater risk of inadequacy during peak flows?</p><p>As demonstrated in previous illustrations (e.g., the industrial park example</p><p>in Sec. 2.1), the expected cost (or benefit) related to the variable is usually</p><p>used to obtain a single number; this cost reflects the sum of all possible</p><p>values of the random cost weighted by the likelihood of their occurrence.</p><p>The use of expected cost in making decisions is the subject of much</p><p>experimental as well as theoretical investigation (Fish-burn [1965]), and it</p><p>will be discussed more fully in Chap. 5. For the time being it can be</p><p>accepted as an intuitively rational description of the economic consequences</p><p>associated with the random variable.</p><p>In general, every value of the random variable leads to a corresponding cost</p><p>or benefit. In other words, we can define cost as a function of the variable:</p><p>images</p><p>Examples of the shapes of such cost functions are sketched in Fig. 2.4.5.</p><p>In many cases the cost function is at least approximately linear. The cost of</p><p>evacuating 100,000 yd3 of earth, for instance, depends linearly on X, the</p><p>price bid per cubic yard, Fig. 2.4.5c. Then C(X) is of the form (in Fig. 2.4.5c</p><p>the constant a is zero)</p><p>images</p><p>and the expected cost</p><p>images</p><p>That is, in this case the expected cost depends only on the mean of the</p><p>random variable.</p><p>images</p><p>Fig. 2.4.5 Some cost functions, (a) Annual rainfall; (b) maximum load on a</p><p>concrete beam; (c) unit bid price.</p><p>In other situations the cost can be approximated by a quadratic relationship.</p><p>Figure 2.4.5a is a possible example. In this case the expected cost depends</p><p>only on the mean and variance of the random variable:</p><p>images</p><p>images</p><p>The importance of the first- and second-order moments, i.e., means and</p><p>variances, of random variables is magnified when it is recognized that</p><p>frequently the ultimate engineering use of the probabilistic model will be in</p><p>a decision-making context where a linear or quadratic cost function is a</p><p>valid approximation. In such cases, the mean and variance are sufficient</p><p>information on which to base the decision if an expected cost criterion is</p><p>used.</p><p>Illustration: Pump-capacity decision This extended illustration is designed</p><p>to demonstrate a number of the concepts discussed to this point in the text</p><p>and to serve as a discussion of others. In particular, a mixed random variable</p><p>will be encountered, a result here of a function g(X) which takes on the same</p><p>value for many different values of X.</p><p>The demand X during the peak summer hour at a water-pumping</p><p>station has a triangular distribution given by</p><p>images</p><p>and sketched in Fig. 2.4.6a.</p><p>The existing pump is adequate for any demand from 0 to 150 cfs. A</p><p>new pump is to be added; it will be operated only if demand exceeds the</p><p>capacity of the existing pump. Hence a demand of up to 150 cfs involves no</p><p>load on the new pump, while a demand of 250 cfs would impose a load of</p><p>100 cfs on the new pump. Let Y be a random variable, “the load on the new</p><p>pump” (i.e., the excess of demand over existing capacity):</p><p>images</p><p>This function is shown in Fig. 2.4.6b. It is not a one-to-one function.</p><p>Let us review how the distribution of Y can be determined. After Eq.</p><p>(2.3.16), we have</p><p>images</p><p>where Ry is the region in which Y = g(X) ≤ y. Y is clearly never</p><p>negative, but it is zero when X is less than 150; thus</p><p>images</p><p>By the symmetry of fX(x) about 150,</p><p>images</p><p>images</p><p>Fig. 2.4.6 Pump-design illustration, (a) Water-demand distribution; (b) load</p><p>on new pump versus demand; (c) CDF of load on new pump; (d) PDF of</p><p>load on new pump.</p><p>Y is never greater than 250 – 150, or 100, but the probability that Y is</p><p>less than any value in the range 0 to 100, say, 30, is the probability that X is</p><p>less than 30 + 150 = 180, or, in general, for 0 ≤ y</p><p>≤ 100,</p><p>images</p><p>Substituting for fX(x) in the range x = 150 to 250,since FX(150) = ½,</p><p>images</p><p>This function is, of course, unity for y > 100 and zero for y 0, written E[Y | Y > 0], is</p><p>images</p><p>This design rule suggests a pump twice as large as the first rule.</p><p>As a less arbitrary and more explicitly cost-oriented approach, the</p><p>engineer might seek that pump capacity z which promises the lowest</p><p>expected total cost. Costs (in arbitrary units) are thought to be described by a</p><p>cost of 100 + 10z (representing a fixed installation cost and the cost of the</p><p>pump itself) plus a cost associated with failing to meet the peak demand.</p><p>Determination of this latter factor is complicated by the possibility of large,</p><p>but off-peak, demands at other times in the same year and by the fact that</p><p>the system will be in place over a number of years. The engineer might</p><p>summarize these effects, however, by saying that they are represented in a</p><p>cost of failing to meet the peak demand in any one arbitrary year given by</p><p>10 + (Y – z)2, if Y – z, the amount by which the pump of capacity z fails to</p><p>meet a random demand of Y, is positive, and zero otherwise. Formally, for a</p><p>given z > 0, the cost as a function of the random variable Y is</p><p>images</p><p>When z = 0, there is no initial cost—only the cost of failing to meet the</p><p>demand.</p><p>For a pump of capacity 0</p><p>or water resource</p><p>facilities. Many questions arise in the practice of using conservative</p><p>estimates. For example:</p><p>How can engineers maintain consistency in their conservatism from one</p><p>situation to another? For instance, separate professional committees set the</p><p>specified minimum concrete compressive strength and the specified</p><p>minimum bending strength of wood.</p><p>Is it always possible to find a value that is conservative in all respects? A</p><p>conservative estimate of the friction factor of a pipe should be on the “high</p><p>side” of the best estimate in order to produce a conservative (low) estimate</p><p>of flow in that pipe, but this estimate may produce unconservative (high)</p><p>estimates of the flows in other parallel pipes in the same network.</p><p>Is the design of the resulting facility unnecessarily expensive? The</p><p>consequences of occasional flows in excess of the capacity of a city storm</p><p>drainage system may be small, but the initial cost of a system capable of</p><p>processing a conservative estimate of the peak flow may be excessive.</p><p>Can the behavior of the facility be adequately predicted with only a</p><p>conservative estimate? For example, the ability of an automatic traffic</p><p>control system to increase the capacity of an artery depends intimately</p><p>upon the degree and character of the variability of the traffic flow as well</p><p>as upon the total volume.</p><p>In short, only if the situation permits can the engineer successfully treat</p><p>uncertainty and variability simply through best estimates or conservative</p><p>estimates. In general, if the decision he must make (i.e., the choice of</p><p>design or course of action) is insensitive to the uncertainty, the engineer</p><p>can ignore it in his analysis. If it cannot be ignored, uncertainty must be</p><p>dealt with explicitly during the engineering process.</p><p>Often, particularly in testing and in laboratory, research, and development</p><p>work, the individual involved may not know how the observed variability</p><p>will affect subsequent engineering decisions. It becomes important, then,</p><p>that he have available efficient ways to transmit his information about the</p><p>uncertainty to the decision makers.</p><p>Although the distinction between these two needs—decision making and</p><p>information transmission in the presence of uncertainty—is not always</p><p>clear, both require special methods of analysis, methods which lie in the</p><p>fields of probability and statistics.</p><p>In Chap. 1 we will review quickly the conventional means for reducing</p><p>observed data to a form in which it can be interpreted, evaluated, and</p><p>efficiently transmitted. Chapter 2 develops a background in probability</p><p>theory, which is necessary to treat uncertainty explicitly in engineering</p><p>problems. Chapter 3 makes use of probability theory to develop a number</p><p>of the more frequently used mathematical models of random processes. In</p><p>Chap. 4 statistical methods that comprise a bridge between observed data</p><p>and mathematical models are introduced. The data is used to estimate</p><p>parameters of the model and to evaluate the validity of the model. Chapters</p><p>5 and 6 develop in detail the methods for analyzing engineering economic</p><p>decisions in the face of uncertainty.</p><p>1</p><p>Data Reduction</p><p>A necessary first step in any engineering situation is an investigation of</p><p>available data to assess the nature and the degree of the uncertainty. An</p><p>unorganized list of numbers representing the outcomes of tests is not easily</p><p>assimilated. There are several methods of organization, presentation, and</p><p>reduction of observed data which facilitate its interpretation and evaluation.</p><p>It should be pointed out explicitly that the treatment of data described in</p><p>this chapter is in no way dependent on the assumptions that there is</p><p>randomness involved or that the data constitute a random sample of some</p><p>mathematical probabilistic model. These are terms which we shall come to</p><p>know and which some readers may have encountered previously. The</p><p>methods here are simply convenient ways to reduce raw data to</p><p>manageable forms.</p><p>In studying the following examples and in doing the suggested problems, it</p><p>is important that the reader appreciate that the data are “real” and that the</p><p>variability or scatter is representative of the magnitudes of variation to be</p><p>expected in civil-engineering problems.</p><p>1.1 GRAPHICAL DISPLAYS</p><p>Histograms A useful first step in the representation of observed data is to</p><p>reduce it to a type of bar chart. Consider, for example, the data presented in</p><p>Table 1.1.1. These numbers represent the live loads observed in a New</p><p>York warehouse. To anticipate the typical, the extreme, and the long-term</p><p>behavior of structural members and footings in such structures, the</p><p>engineer must understand the nature of load distribution. Load variability</p><p>will, for example, influence relative settlements of the column footings.</p><p>The values vary from 0 to 229.5 pounds per square foot (psf). Let us divide</p><p>this range into 20-psf intervals, 0 to 19.9, 20.0 to 39.9, etc., and tally the</p><p>number of occurrences in each interval.</p><p>Plotting the frequency of occurrences in each interval as a bar yields a</p><p>histogram, as shown in Fig. 1.1.1. The height, and more usefully, the area,</p><p>of each bar are proportional to the number of occurrences in that interval.</p><p>The plot, unlike the array of numbers, gives the investigator an immediate</p><p>impression of the range of the data, its most frequently occurring values,</p><p>and the degree to which it is scattered about the central or typical values.</p><p>We shall learn in Chap. 2 how the engineer can predict analytically from</p><p>this shape the corresponding curve for the total load on a column</p><p>supporting, say, 20 such bays.</p><p>Table 1.1.1 Floor-load data*</p><p>images</p><p>* Observed live loads (in pounds per square foot); bay size: 400 ± 6 ft2.</p><p>Source: J. W. Dunham, G. N. Brekke, and G. N. Thompson [1952], “Live</p><p>Loads on Floors in Buildings,” Building Materials and Structures Report</p><p>133, National Bureau of Standards, p. 22.</p><p>images</p><p>Fig. 1.1.1 Histogram and frequency distribution of floor-load data.</p><p>If the scale of the ordinate of the histogram is divided by the total number</p><p>of data entries, an alternate form, called the frequency distribution, results.</p><p>In Fig. 1.1.1, the numbers on the right-hand scale were obtained by</p><p>dividing the left-hand scale values by 220, the total number of</p><p>observations. One can say, for example, that the proportion of loads</p><p>observed to lie between 120 and 139.9 psf was 0.10. If this scale were</p><p>divided by the interval length (20 psf), & frequency density distribution</p><p>would result, with ordinate units of “frequency per psf.” The area under this</p><p>histogram would be unity. This form is preferred when different sets of</p><p>data, perhaps with different interval lengths, are to be compared with one</p><p>another.</p><p>The cumulative frequency distribution, another useful graphical</p><p>representation of data, is obtained from the frequency distribution by</p><p>calculating the successive partial sums of frequencies up to each interval</p><p>division point. These points are then plotted and connected by straight lines</p><p>to form a nondecreasing (or monotonic) function from zero to unity.</p><p>In Fig. 1.1.2, the cumulative frequency distribution of the floor-load data,</p><p>the values of the function at 20, 40, and 60 psf were found by forming the</p><p>partial sums 0 + 0.0455 = 0.0455, 0.0455 + 0.0775 = 0.1230, and 0.1230 +</p><p>0.1860 = 0.3090.† From this plot, one can read that the proportion of the</p><p>loads observed to be equal to or less than 139.9 psf was 0.847. After a</p><p>proper balancing of initial costs, consequences of poor performance, and</p><p>these frequencies, the designer might conclude that a beam supporting one</p><p>of these bays must be stiff enough to avoid deflections in excess of 1 in. in</p><p>99 percent of all bays. Thus the design should be checked for deflections</p><p>under a load of 220 psf.</p><p>Some care should be taken in choosing the width of each interval in these</p><p>diagrams, † A little experimentation with typical sets of data will convince</p><p>the reader that the choice of the number of class intervals can alter one’s</p><p>impression of the data’s behavior a great deal. Figure 1.1.3</p><p>value of X. Notice that this may</p><p>be written as</p><p>images</p><p>The second integral is simply the marginal distribution fX(x) [Eq. (2.2.43)];</p><p>thus</p><p>images</p><p>which is the same definition given in Eq. (2.4.2). Thus the expected value of</p><p>X, Eq. (2.4.53), is the average value of X “without regard for the value of Y.”</p><p>The definition and meaning of E[Y] are, of course, similar.</p><p>It may be helpful for the civil engineer to observe that mX and mY locate the</p><p>center of mass of a two-dimensional plate of variable density or the</p><p>horizontal coordinates of the center of mass of the “hill” or terrain whose</p><p>surface elevations are given by fX,Y(x,y). This follows from the definition of</p><p>these terms and the fact that the volume under fX,Y(x,y) is unity.</p><p>By the same analogy, the second-order moments (l = 2, n = 0), (l = 0, n = 2),</p><p>and (l = 1, n = 1) correspond respectively to the moment of inertia about the</p><p>x axis, the moment of inertia about the y axis, and the product moment of</p><p>inertia with respect to these axes.</p><p>Central moments As for a single variable, and as in mechanics, the more</p><p>useful second (and higher) moments are those with respect to axes passing</p><p>through the center of mass. Taking g(X,Y) equal to</p><p>images</p><p>we find</p><p>images</p><p>which is called a central moment of order l + n of the random variables X</p><p>and Y. The first central moments are, of course, both zero.</p><p>The most valuable central moments are the set of second-order moments: (l</p><p>= 2, n = 0), (l = 0, n = 2), and (l=1, n = 1). The first two cases, as above</p><p>with the means, reduce to the marginal variances. With (l = 2, n = 0), for</p><p>example,</p><p>images</p><p>The result for Y, (l = 0, n = 2), is similar.</p><p>Covariance The new type of second central moment that is found when</p><p>joint random variables are considered is that involving their product, that is,</p><p>(l = 1, n = 1). This moment also has a name, the covariance of X and Y: Cov</p><p>[X,Y] or† σX,Y.</p><p>images</p><p>If the variances correspond to the moments of inertia about axes in the x and</p><p>y direction passing through the centroid of a thin plate of variable density,</p><p>then the covariance corresponds to its product moment of inertia with</p><p>respect to these axes.</p><p>Correlation coefficient A normalized version of the covariance, called the</p><p>correlation coefficient ρX,Y,, is found by dividing the covariance of X and Y</p><p>by the product of their standard deviations:</p><p>images</p><p>It can be shown that this coefficient has the interesting property that</p><p>images</p><p>Before discussing the interpretations of the covariance and the correlation</p><p>coefficient (and its bounds, ±1), let us illustrate the computation of these</p><p>numbers in a simple discrete case. Consider the discrete joint distribution of</p><p>X and Y sketched in Fig. 2.4.8, where the distribution might represent a</p><p>discrete model of the maximum annual flows at gauge points in two</p><p>different, but neighboring, streams. The engineer’s interest in their joint</p><p>behavior might arise from his concern over flooding of the river which the</p><p>streams feed or from his desire to estimate flow in one stream by measuring</p><p>only the flow in the other.</p><p>images</p><p>Fig. 2.4.8 Joint PMF of stream flows.</p><p>The means and variances are found from the marginal PMF’s, or as follows:</p><p>images</p><p>images</p><p>images</p><p>Similarly</p><p>images</p><p>images</p><p>images</p><p>The covariance is found using the discrete analog of Eq. (2.4.58) or</p><p>images</p><p>which here becomes</p><p>images</p><p>The correlation coefficient pX,Y is</p><p>images</p><p>images</p><p>Fig. 2.4.9 Joint density-function contours of correlated random variables. (a)</p><p>Positive correlation ρ > 0; (b) high positive correlation ρ ≈ 1; (c) negative</p><p>correlation ρ</p><p>(for example, Y =</p><p>aX2) may yield a small correlation coefficient. Less than functional</p><p>dependence but a close (nonlinear) relationship such as indicated in Fig.</p><p>2.4.9e, can also yield a small or zero correlation coefficient. Stochastic</p><p>dependence in such a case is clearly large; given a value of X = x, the</p><p>conditional distribution of Y will greatly differ from its marginal</p><p>distribution. But if one mistakenly concludes a lack of strong stochastic</p><p>dependence from a small value of the correlation coefficient, he will not</p><p>reach the proper conclusion in such cases.</p><p>In summary, given only the value of the correlation coefficient, one can</p><p>conclude from a high value that the stochastic dependence is high and</p><p>furthermore that X and Y have a joint linear tendency, whereas a small value</p><p>implies only the weakness of a linear trend and not necessarily a weakness</p><p>in stochastic dependence.</p><p>The definitions of moments may be generalized in a straightforward manner</p><p>to more than two jointly distributed random variables, but the use of higher</p><p>than second-order moments is seldom encountered in practice. Further</p><p>discussion of moments will be postponed until they are needed.</p><p>Properties of expectation As suggested, Eq. (2.4.50) provides a more</p><p>efficient way of computing the expected value of a random variable, Z =</p><p>g(X,Y), than that of first finding its probability law and then finding its</p><p>expectation. That is, if, in the discrete example of stream flows above, the</p><p>mean of the total at the maximum flows ‡ Z = X +Y, is desired, it could be</p><p>found by finding pZ (z) and then averaging. Alternatively and more simply,</p><p>using Eq. (2.4.50), E[Z]</p><p>images</p><p>Even more simply we can make use of the easily verified linearity property</p><p>of expectation</p><p>images</p><p>For example, in this special case,</p><p>images</p><p>The linearity property can be used, as in the preceeding section, to find</p><p>relationships among expectations which simplify computations, or which,</p><p>because they are independent of the underlying distributions, are valuable</p><p>when only these expectations are known.</p><p>For example, the following relationship often simplifies covariance</p><p>computation</p><p>images</p><p>which, using the linearity property, reduces to</p><p>images</p><p>Illustration: Correlated demand and capacity We illustrate the use of the</p><p>linearity property in situations where moments are known, but distributions</p><p>are not, by the following example. If the capacity C of an engineering</p><p>system and the demand upon it D are random variables, the margin M = C –</p><p>D is a measure of the performance of the system, whether it be inadequate,</p><p>M</p><p>will show that</p><p>images</p><p>and that</p><p>images</p><p>Illustration: Correlation versus functional dependence The correlation</p><p>coefficient alone is frequently used to make deductions about the joint</p><p>behavior of random variables. In many fields, particularly the biological and</p><p>social sciences, where basic laws are difficult to derive otherwise, it is</p><p>common to measure repeatedly two variables which are suspected or</p><p>hypothesized to be related and then to estimate † their correlation</p><p>coefficient. If the coefficient is near unity in absolute value, strong</p><p>dependence is assumed verified; if the coefficient is low, it is concluded that</p><p>one variable has little or no effect on the other.</p><p>Such techniques of verification or determination of suspected</p><p>relationships are becoming more and more common in civil engineering,</p><p>where many of our common materials—e.g., concrete and soil—and many</p><p>of our systems—e.g., watersheds and traffic—are so complex that</p><p>relationships among variables can only be determined empirically. To use</p><p>such an approach correctly, it is necessary to have a sound understanding of</p><p>its implications and its potential errors. The following extended illustration</p><p>is designed both as an exercise in using expectation operations and as an aid</p><p>to understanding the effects on the correlation coefficient caused by such</p><p>factors as the character of the functional relationship and the presence of</p><p>other causes of variation. It is important to appreciate the fact that the</p><p>conclusions deduced from this study do not depend on the shape of the joint</p><p>probability distribution. This fact demonstrates the power and generality of</p><p>working with means, variances, and correlations, without detailed</p><p>probability law specifications.</p><p>Suppose that an engineer is investigating the possible relationship</p><p>between density of a particular highway subbase material and the road’s</p><p>performance. In each of a number of segments of a road of nominally</p><p>uniform design, he measures both the density of the subbase material (found</p><p>in a core drilled through the pavement) and the value of some index of the</p><p>segment’s performance, namely, the smoothness of the ride provided to</p><p>passing vehicles. ‡ Let the random variable Y be the “ridability” or</p><p>performance index and let X be the deviation of the subbase density from the</p><p>specified (say, the mean) density. §</p><p>The engineer could estimate from the data the correlation coefficient,</p><p>ρX,Y. Generally, a low value of this coefficient would lead him to conclude</p><p>that performance (ridability) is virtually independent of subbase density. A</p><p>relatively high value of ρX,Y, on the other hand, would normally cause the</p><p>conclusion that subbase density strongly affects performance (and hence,</p><p>perhaps, must be well controlled in future jobs, if high performance is to be</p><p>obtained).</p><p>In order to understand the general validity of such conclusions, we</p><p>want to presume that, in fact, a physical law or function (unknown to the</p><p>engineer, of course) governs the relationship between performance Y and</p><p>subbase density X. The law we shall assume is of the form</p><p>images</p><p>in which a, b, and c are constants and in which W is a random variable</p><p>with mean zero, stochastically independent of X, and W represents the</p><p>variation in Y attributable to factors other than the subbase density X. These</p><p>might include such factors as settlement of the soil fill below the subbase or</p><p>variations in pavement placement. Assuming the average effect of these</p><p>factors is accounted for in the constant a, W need only represent the effect</p><p>on Y of the random variations about the mean values of these factors. Thus</p><p>the mean zero assumption is reasonable.</p><p>Calculation of the Correlation Coefficient First, as an exercise in dealing</p><p>with expectation we seek the correlation coefficient ρX,Y given that Eq.</p><p>(2.4.89) defines the relationship between Y, X, and the other factors. From</p><p>Eq. (2.4.72),</p><p>images</p><p>Evaluating the first term with the use of the linearity property,</p><p>images</p><p>Owing to the assumed independence of W and X, E[WX] = E[W]E[X].</p><p>Since E[W] was assumed to be zero, the last term is zero. Assuming that X,</p><p>the deviation from the specified subbase density, is symmetrically</p><p>distributed about zero,† E[X] = 0, E[X3] = 0 and Var [X] = E[X2]. Thus</p><p>images</p><p>The sign of the covariance in this example depends only on the sign of</p><p>b.</p><p>To find the correlation coefficient, we need the standard deviations</p><p>of X and Y:</p><p>images</p><p>Owing to the assumptions above, many terms drop from this</p><p>expression. Combining these results and grouping terms,</p><p>images</p><p>Simply to shorten the expression, we substitute† Var [X2] for E[X4] –</p><p>E2[X2]:</p><p>images</p><p>Finally,</p><p>images</p><p>Let us concentrate on ρ 2, since it is simpler in form than ρ. Of course ρ</p><p>= 0 implies ρ2 = 0 and ρ = ±1 implies ρ2 = 1; hence the same implications</p><p>regarding presence or lack of dependence follow from ρ2 exactly as from ρ.</p><p>Simplifying,</p><p>images</p><p>Note that no use has been made of integration or density functions in</p><p>obtaining these results.</p><p>Values of ρ Implied by the Form of the True Law We are now prepared to use</p><p>this equation to demonstrate how various conditions in the true underlying</p><p>law, Eq. (2.4.89), affect the value of the correlation coefficient and hence the</p><p>validity of conclusions drawn from its value. The joint density functions</p><p>implied by each case are sketched in Fig. 2.4.10.</p><p>Case 1: Y = a + bX Notice first that the only way a value of ρ (or ρ2)</p><p>equal to unity will be found is if b is not equal to zero but both c and σW</p><p>2</p><p>are, that is, if a linear functional relationship, ‡ Y = a + bX, and hence</p><p>perfect stochastic dependence, exists. Values of ρ2 less than 1 will be found</p><p>in all other situations.</p><p>Case 2: Y = a + bX + W Let us look next at the case when c = 0, that</p><p>is, when a linear relationship between X and Y exists, but other sources of</p><p>randomness Y are present:</p><p>images</p><p>Then</p><p>images</p><p>In this case, if the variance of the other contributions to the variation in</p><p>Y is relatively small, the correlation coefficient will be very nearly unity. The</p><p>engineer’s conclusions, based on observing ρ ≈ 1, that strong stochastic</p><p>dependence (almost a simple functional dependence) exists between X and Y</p><p>would be valid. If, however, the contributions to the scatter in Y come</p><p>predominantly from other factors, and/or if the functional dependence</p><p>between X and Y is weak (b is small), then images As a result the</p><p>correlation coefficient will be very small. A conclusion (based on observing</p><p>a small value of ρ) that there is no relationship between X and Y would not</p><p>be valid, as a linear relationship does exist. But ρ does remain a good</p><p>indicator of the stochastic dependence between X and Y, for this will be</p><p>small too; the conditional distribution of Y given X = x will have variance</p><p>σw</p><p>2, very nearly as large as that of marginal distribution of Y, which is b2σX</p><p>2</p><p>+ σw</p><p>2. In this case the random variation W, which is the result of other</p><p>factors, masks the linear Y verses X relationship. The implication for field</p><p>practice and betterment of performance is simply that, even though there</p><p>may be a functional relationship between performance and subbase density,</p><p>it is so weak or the variance of subbase density is so small that, compared</p><p>with the influence of the other factors, the deviations in subbase density</p><p>contribute little to the variations in performance. Hence the conclusion,</p><p>based on observation of a small value of ρ2, that for nominally similar</p><p>designs (i.e., same mean density) there is little effect of subgrade density on</p><p>performance may be a valid one operationally.</p><p>images</p><p>Fig. 2.4.10 Joint density-function contours fX,Y(x,y) for various cases of a</p><p>physical law governing Y and X. In cases 1 and 3, the PDF’s degenerate into</p><p>“walls” which are only suggested graphically.</p><p>Case 3: Y = a + bX + cX2 Next we look at another limiting</p><p>case,</p><p>namely when σw</p><p>2 = 0, that is, when the variation in other factors is</p><p>negligible or at least has negligible effect upon performance. Now a</p><p>functional relationship between X and Y exists:</p><p>images</p><p>and so does perfect stochastic dependence; i.e., the conditional</p><p>distribution of Y given X = x is a spike of unit mass at the value a + bx + cx2.</p><p>But the correlation coefficient may well prove misleading:</p><p>images</p><p>If the squared term dominates over the linear one, that is, if images</p><p>the correlation coefficient will be small even though dependence is perfect.</p><p>In this situation an engineer’s conclusion, based on an observed small value</p><p>of ρ2, that subgrade density does not significantly affect performance, would</p><p>be incorrect. Subsequent action based on neglecting the existing dependence</p><p>could lead to relaxing of quality control standards on subgrade construction</p><p>and consequent deterioration in performance (assuming b and c are</p><p>negative). In this case a plot of the data points will usually exhibit the strong</p><p>functional relationship which the correlation coefficient fails to suggest.</p><p>Note, on the other hand, that if the linear term predominates, i.e., if images</p><p>, the correlation will be high (but never unity) and will properly indicate</p><p>strong dependence. This illustrates why it was stated that the correlation</p><p>coefficient should properly be called a measure of linear dependence.† It</p><p>must be pointed out that these last conclusions from the illustration depend</p><p>on the assumption that E[X3] =0.† If this condition does not hold, the</p><p>correlation coefficient is generally enhanced as a measure of dependence.</p><p>No matter what the relative values of c and b, if additional sources of</p><p>variation W also exist, they will tend to decrease ρ2 [Eq. (2.4.94)] and mask</p><p>further the functional relationship between X and Y.</p><p>Spurious Correlation It is important, too, to point out another potential</p><p>source of misinterpretation associated with the use of correlation studies.</p><p>The discussion above revealed that small correlation does not necessarily</p><p>imply weak dependence, and it is also true that spurious correlation may</p><p>invite unwarranted conclusions of a cause and effect relationship between</p><p>variables when, in fact, none exists. In the example here, for instance, some</p><p>other variable factor, such as moisture content of the soil or volume of heavy</p><p>trucks, might cause variation in both subbase density and performance.</p><p>Hence higher-than-average values of density might, in fact, usually be</p><p>paired with higher-than-average performance (and low with low), yielding a</p><p>positive correlation that is not a result of subbase density’s beneficial effect</p><p>on performance but is rather a result of both factors being increased (or</p><p>decreased) by presence of the high (or low) value of the third factor. This</p><p>fact does not, however, reduce the value of using an observation of one</p><p>variable to help predict the other, as long as the factor causing the</p><p>correlation is not altered.</p><p>Benson [1965] reported on spurious correlation and on its potential</p><p>and actual presence in many civil-engineering investigations, even those</p><p>which are deterministic rather than probabilistic in formulation. He cites as a</p><p>particularly common source of spurious correlation that introduced by the</p><p>engineer who seeks to normalize his data by dividing it by a factor which is</p><p>itself a random variable. For example, correlation between X, accidents, and</p><p>Y, out-of-town drivers, is not necessarily implied by correlation (observed in</p><p>a number of cities) between X/Z, the number of accidents per registered</p><p>vehicle, and Y/Z, the number of commuters per registered vehicle. Formally,</p><p>X and Y may be independent, but X/Z and Y/Z are not, and high correlation</p><p>between the latter pair does not imply correlation between the former pair.</p><p>Similarly, correlation between live load per square foot and cost per square</p><p>foot in structures need not imply that cost and load are dependent, because</p><p>both may depend upon the normalizing factor, that is, total floor area.</p><p>Summary The correlation coefficient provides a very useful measure of</p><p>dependence between variables, but the engineer must be careful in its</p><p>interpretation. Large values of p2 imply strong stochastic dependence and a</p><p>near-linear functional relationship, but not necessarily a cause-and-effect</p><p>relationship. Small values may result from the predominance of other</p><p>sources of variation (and hence low stochastic dependence), or, if a</p><p>functional relationship exists, from its lack of strong linearity.</p><p>Conditional expectation and prediction† As with single random variables</p><p>there is often advantage in working with expected values conditioned on</p><p>some event A. In general we have</p><p>images</p><p>in which fX,Y|[A](x,y) is the joint conditional distribution of X and Y given the</p><p>event A [for example, Eq. (2.2.58)].</p><p>The single most important application of conditional expectation is in</p><p>prediction. If Y is a maximum stream flow or the peak afternoon traffic</p><p>demand on a bridge, the engineer may be asked to “predict what Y will be.”</p><p>To “predict Y” is to state a single number, which, in some sense, is the best</p><p>prediction of what value the random variable Y will take on. Without</p><p>additional information, the (marginal) mean of Y, or mY, is conventionally</p><p>used to predict Y. Intuitively it is a reasonable choice. More formally it is the</p><p>predictor of Y which has a minimum expected squared error (or “mean</p><p>square error”). The error in predicting Y by mY is simply Y – mY. Its</p><p>expected square is</p><p>images</p><p>or the (marginal) variance of Y. No other predictor will give a smaller mean</p><p>square error. † It should be emphasized that the mean square error criterion</p><p>for evaluating a predictor is simply a mathematically convenient engineering</p><p>convention. More generally the engineer should use a predictor which</p><p>minimizes the expected cost, the cost being some function of the error. If</p><p>this cost is approximately proportional to the square of the error, whether</p><p>positive or negative, then the mean square error criterion is also a minimum</p><p>expected cost criterion and the mean is the best predictor. If, on the other</p><p>hand, there are larger costs associated with underestimating Y than with</p><p>overestimating it, the minimum cost predictor will be greater than the mean.</p><p>Such economic questions are more properly treated in Chaps. 5 and 6; we</p><p>restrict attention here to the conventional mean square error criterion.</p><p>Suppose now that the engineer learns that another random variable X has</p><p>been observed to be some particular value, say x. The hydraulic engineer</p><p>may have learned the maximum flow in a neighboring river or at an</p><p>upstream point, or the traffic engineer may have learned that the morning</p><p>peak flow was x = 1000 cars per hour. Conditional on this new information,</p><p>what value should the engineer use to predict Y? Applying the same</p><p>argument used above to the conditional distribution of Y given that X = x,</p><p>one concludes that the “best” predictor of F is the conditional mean, or mY|X.</p><p>images</p><p>images</p><p>Fig. 2.4.11 Illustration of conditional means and variances of Y given X.</p><p>Note that in general this conditional mean † depends on the value x. This</p><p>predictor has a mean square error equal to the conditional variance of Y</p><p>given X = x:</p><p>images</p><p>This variance is also a function of x, in general. These relationships are</p><p>sketched in Fig. 2.4.11, where the square root of the mean square error (or</p><p>rms) is indicated rather than the mean square error itself.</p><p>In Sec. 3.6.2 we will discuss in detail an illustration of conditional prediction</p><p>applied to a particular, commonly adopted, joint distribution of X and Y. In</p><p>that case we will find, upon carrying out the integrations indicated above,</p><p>that</p><p>images</p><p>and that</p><p>images</p><p>We note in this special application that the predictor mY|X is linear in x and</p><p>that the mean square error is a constant, independent of x. The importance,</p><p>once again, of the correlation coefficient ρ is observed. For this particular</p><p>joint distribution, † given X = x, the</p><p>amount the conditional predictor mY|X</p><p>will be altered from the (marginal) predictor mY is directly proportional to ρ.</p><p>Also, the uncertainty in predicting Y given X = x (as measured by the mean</p><p>square error or conditional variance) is smaller than the marginal uncertainty</p><p>σY</p><p>2 by an amount proportional to the square of ρ.</p><p>Illustration: Sum of a random number of random variables Conditional</p><p>expectation is also valuable when studying probability models. Often</p><p>moments of important variables may be difficult to determine unless one</p><p>takes advantage of conditional arguments.</p><p>For example, recall that in Sec. 2.3.3 we assumed that T, the total</p><p>rainfall on a watershed, was the sum of the (independent, identically</p><p>distributed) random rainfalls, R1, R2, . . ., in N “rainy” weeks, where the</p><p>number of rainy weeks is itself random (and independent of the Ri’s):</p><p>images</p><p>The estimation of the distribution of T by simulation was found to be a</p><p>long numerical task. In addition, it required complete knowledge of the</p><p>distribution of N and Ri. Having knowledge only of the means and variances</p><p>of N and Ri, it would be desirable to be able to calculate explicitly the mean</p><p>and variance of T. They are</p><p>images</p><p>and</p><p>images</p><p>The randomness in N makes a direct attack on this problem</p><p>troublesome. But, given the number of rainy weeks (that is, conditional on N</p><p>= n), we can apply familiar formulas [Eqs. (2.4.81a) and (2.4.81b)] for the</p><p>sum of a known number of (independent) random variables:</p><p>images</p><p>and</p><p>images</p><p>in which σR</p><p>2 and mR are the variance and mean of any Ri. The subscript</p><p>R on the expectation symbol is simply a reminder that this expectation is</p><p>with respect to the R’s. Now, recognizing that N is in fact random, we have</p><p>two simple functions of the random variable N:</p><p>images</p><p>To find E[T] and E[T2], we need simply take expectation of g1(N) and</p><p>g2(N) with respect to N:</p><p>images</p><p>and</p><p>images</p><p>This implies</p><p>images</p><p>In words, the expected value of the sum of a random number of random</p><p>variables is just the mean number times the mean of each variable. The</p><p>variance of the sum is, however, larger than the mean number times the</p><p>variance of each variable (as it would be if N were not random but known to</p><p>equal its mean). The additional term is just the variance of the mean of each</p><p>variable times N; that is,</p><p>images</p><p>In short,</p><p>images</p><p>This result is intuitively satisfying but hardly easy to anticipate. It</p><p>shows clearly the potential danger (or error) in oversimplifying probabilistic</p><p>models. If the engineer had tried, in this case, to simplify his model either by</p><p>replacing the number of rainy weeks by their average number or by</p><p>replacing the rainfall in a rainy week by its mean value, he would have</p><p>underestimated the uncertainty (variance) in the total rainfall. If, however,</p><p>either σN</p><p>2 or σR</p><p>2 were small enough, the error induced by the corresponding</p><p>simplification would not be significant.</p><p>It is important to appreciate that again we have been able to analyze</p><p>a probability model using only means and variances. No assumptions were</p><p>made in this example about the shapes of the distributions of R and N.</p><p>Expectations alone were used, and this was done symbolically (or</p><p>operationally) without resort to formal integration.</p><p>2.4.4 Approximate Moments and Distributions of Functions†</p><p>Approximate solutions to the problem of determining the behavior of</p><p>dependent random variables are usually possible. The approximations have</p><p>the advantage of always giving moments of dependent variables only in</p><p>terms of functions of moments of the independent variables. As stressed</p><p>before, these moments may be all that are available or all that are necessary</p><p>for the engineer’s purposes. Simple approximations to the distribution of</p><p>dependent random variables are also possible.</p><p>Approximate moments: Y = g(X) If the relationship Y = g(X) is sufficiently</p><p>well behaved, and if the coefficient of variation of X is not large, ‡ the</p><p>following approximations are valid</p><p>images</p><p>images</p><p>or</p><p>images</p><p>where dg(x)/dx|mx signifies the derivative of g(x) with respect to x, evaluated</p><p>at mX.</p><p>If, for example, Y = a + bX + cX2, then the approximations state that</p><p>images</p><p>images</p><p>or</p><p>images</p><p>That the approximations are not exact can be observed by looking at the last</p><p>term in Eq. (2.4.109) which should be cE[X2] or c(mX</p><p>2 + σX</p><p>2) or cmX</p><p>2(l +</p><p>VX</p><p>2). Clearly if the coefficient of variation of X is less than 10 percent, the</p><p>error involved in this approximation is less than 1 percent.</p><p>The justification for these approximations lies in the observation that if VX is</p><p>small, X is very likely to lie close to mX and hence a Taylor-series expansion</p><p>of g(X) about mX is suggested:</p><p>images</p><p>Keeping only the first two terms in the expansion and taking the expectation</p><p>of both sides, we obtain the stated approximation for E[g(X)], since E[X –</p><p>mX] = 0. Similarly, keeping the same terms and finding the variance of both</p><p>sides yields the approximation in Eq. (2.4.107), since</p><p>images</p><p>and</p><p>images</p><p>It is of course possible to increase accuracy by keeping additional terms in</p><p>the expansion and by involving higher moments in the approximations. The</p><p>mean is better approximated in general by using three terms of the</p><p>expansion when taking expectations yields</p><p>images</p><p>Notice that in the quadratic case, Y = a + bX + cX2, this “approximation” is</p><p>exact:</p><p>images</p><p>This is true, of course, because the higher-order derivatives and hence the</p><p>higher-order terms of the Taylor-series expansion are all zero. In practice</p><p>this second-order approximation of the mean [Eq. (2.4.113)] and the first-</p><p>order approximation of the variance and standard deviation [Eqs. (2.4.107)</p><p>and (2.4.108)] are commonly used, since they depend upon only the mean</p><p>and variance of X.</p><p>Approximate distribution: Y = g(X) Similar reasoning can be used to find</p><p>an approximate distribution for Y (at least in the region of the mean) from</p><p>the Taylor expansion of Y = g(X). Keeping only the first two or the linear</p><p>terms in X, we can write</p><p>images</p><p>which is of the linear form</p><p>images</p><p>Applying Eq. (2.3.15), for a continuous distribution of X,</p><p>images</p><p>The implication is that the approximate fY(y) has the same shape as fX(x),</p><p>only stretched and shifted. In essence the technique replaces the true</p><p>relationship between X and Y by the linear one associated with the tangent to</p><p>g(X) at mX. Consequently one can expect that the approximate distribution</p><p>will be reasonably close to the exact one at those values of y where the</p><p>tangent provides a reasonable approximation to g(X). If, owing to the shape</p><p>of fX(x), X is confined to lie in this same region (at least with high</p><p>likelihood), the indicated distribution of Y will be a good approximation to</p><p>the true fY(y) almost everywhere. A quick sketch of fX(x) and g(X) should</p><p>determine if these conditions hold.</p><p>For illustration assume that Y = X2 and find the approximate distribution of</p><p>fY(y) when X has the distribution</p><p>images</p><p>with mX = 5, by inspection. Then</p><p>images</p><p>and</p><p>images</p><p>and</p><p>images</p><p>Hence</p><p>images</p><p>Notice that the limits on the definition also must come from the approximate</p><p>relationship, e.g., from y ≥ –25 + 10(4) = 15, not from y ≥ 42 = 16 to</p><p>guarantee that the resulting distribution is a proper one. The true</p><p>distribution, using Eq. (2.3.4), † is</p><p>images</p><p>This situation is sketched in Fig. 2.4.12, in a manner analogous to that used</p><p>for Fig. 2.3.4. There is also shown in the figure the case where fX(x) is</p><p>images in the region 0 ≤ x ≤ 2, a region in which the nonlinearity of g(x) is</p><p>more severe and where the approximate distribution is therefore less precise.</p><p>images</p><p>Fig. 2.4.12 Approximate distributions</p><p>Multivariate approximations Similar approximations can be made in</p><p>multivariate situations. Now a multidimensional Taylor-series expansion is</p><p>necessary before expectations are taken.</p><p>The second-order approximation to the expected value of</p><p>images</p><p>is</p><p>images</p><p>in which (∂2g/∂xi ∂xj)|m is the mixed second partial derivative of g(x1,x2, . . .</p><p>,xn) with respect to xi and xj evaluated at images. The second term (which</p><p>reduces to one-half the sum of the variances times the second derivatives if</p><p>the Xi are uncorrelated) is negligible if the coefficients of variations of the Xi</p><p>and the nonlinearity in the function are not large.</p><p>The first-order approximation to the variance of Y is</p><p>images</p><p>which, if the Xi are uncorrelated, is simply</p><p>images</p><p>Notice the form of this equation. It may be interpreted as meaning that each</p><p>of the n random variables Xi contributes to the dispersion of Y in a manner</p><p>proportional to its own variance Var [Xi] and proportional to a factor [(∂g/</p><p>∂xi) |m]2, which is related to the sensitivity of changes in Y to changes in X.</p><p>We can use this interpretation to evaluate quantitatively the common</p><p>practice of using engineering judgement to simplify problems. In many</p><p>cases it is sufficiently accurate to treat some of the independent variables as</p><p>deterministic rather than as stochastic. This formula indicates that the effect</p><p>of this action is to neglect a contribution to the variance of Y and that the</p><p>simplification is a justified approximation if either the variance or the</p><p>“sensitivity” factor of that variable is small enough that their product is</p><p>negligible compared to the contributions of other variables in the problem.</p><p>In the study of the variance of the ultimate moment capacity of reinforced-</p><p>concrete beams, for example, it is found that the “most uncertain”</p><p>independent variable,† namely, the ultimate concrete strength, can in</p><p>practice be treated as a constant because, owing to a small “sensitivity”</p><p>factor, it contributes relatively little to the variance of the ultimate moment.</p><p>For similar reasons it is reasonable to treat the width of the beam as</p><p>deterministic, whereas the depth to the steel reinforcement makes a major</p><p>contribution to variation in the ultimate moment.</p><p>To complete the spectrum of second-order moment approximations we need</p><p>the covariance between two functions, Y1 and Y2, of the X’s:</p><p>images</p><p>images</p><p>In this case,</p><p>images</p><p>It should be pointed out that, again, first- and second-order moments are</p><p>sufficient to provide at least approximations of the same moments of</p><p>functionally dependent random variables.</p><p>For example, if X1 through Xn are mutually independent, we have,</p><p>generalizing Eq. (2.4.86), product</p><p>images</p><p>The answer based on the approximation technique is the same. Exact</p><p>expressions for variances are more cumbersome (although possible), but the</p><p>approximate result is simply</p><p>images</p><p>For n = 2,</p><p>images</p><p>Compare this result with the exact result, Eq. (2.4.87), which has an added</p><p>term images, which is relatively small if the coefficients of variation are</p><p>small. Notice that, approximately,</p><p>images</p><p>This useful result, true in general for two or more variables, says that the</p><p>square of the coefficient of variation of the product of uncorrelated random</p><p>variables is approximately equal to the sum of the squares of the coefficients</p><p>of variation of the variables. Compare this with the analogous statement for</p><p>the variance of the sum of uncorrelated random variables [Eq. (2.4.81c)].</p><p>The previous results state, for illustration, that the mean force F on a body</p><p>submerged in a moving fluid (a truss bar in the wind or a grillage in a stream</p><p>of water, for example) is</p><p>images</p><p>when F = RCAS2. Here R is the density of the fluid, S is the velocity of the</p><p>fluid, C is the body’s shape-dependent, empirical drag coefficient, A is its</p><p>exposed area, and the variables are assumed mutually independent. The</p><p>coefficient of variation of the force is approximately</p><p>images</p><p>The coefficient of variation of the square of S is, by definition,</p><p>images</p><p>which, in the light of Eqs. (2.4.106) and (2.4.108), is, approximately,</p><p>images</p><p>In practice, one or more of these coefficients might be sufficiently small that</p><p>the corresponding variable could be treated as a deterministic constant rather</p><p>than as a random variable.</p><p>Approximate distributions of functions of jointly distributed random</p><p>variables can also often be found more easily than the exact distribution by</p><p>using the linear part of a Taylor-series expansion and the relatively simple</p><p>relationships † for the distributions of linear functions of random variables.</p><p>The procedure is straightforward; no further discussion will be given here.</p><p>2.4.5 Summary</p><p>This section presents in detail methods of analysis of probabilistic models</p><p>based on moments rather than on entire distributions. Defining the</p><p>expectation E[g(X)] of a function g(X) of a random variable X as a weighted</p><p>average, that is,</p><p>images</p><p>we recognize that the two most important cases are first and second</p><p>moments, the mean of a random variable</p><p>images</p><p>and the variance</p><p>images</p><p>The standard deviation σx is the positive square root of the variance, and the</p><p>coefficient of variation VX is the ratio σX/mX.</p><p>These formulas show that these expectations can be calculated by</p><p>integration (or summation) if the probability law ƒx(x) is known. More</p><p>important is the fact that even without knowledge of complete distributions,</p><p>useful practical analysis of engineering problems can be carried out with</p><p>expectations alone. Certain rules of thumb and inequalities permit</p><p>probabilities to be related to these moments. Economic analyses can be</p><p>based on expected costs. Owing primarily to the linearity property of the</p><p>expectations operation,</p><p>images</p><p>moment analysis of models can often be completed with relative ease and</p><p>without explicit use of integration.</p><p>In particular, this property leads to important formulas for the mean and</p><p>variance of a sum of random variables. For two variables,</p><p>images</p><p>The covariance is defined as</p><p>images</p><p>The correlation coefficient ρ is defined as</p><p>images</p><p>The latter coefficient is bounded between –1 and +1 and is a useful measure</p><p>of linear dependence when properly interpreted.</p><p>The concept of conditional expectation and the availability of approximate</p><p>expressions for moments (Sec. 2.4.4) extend the range of probabilistic</p><p>models to which moment analysis can be applied without complete</p><p>distribution information.</p><p>2.5 SUMMARY FOR CHAPTER 2</p><p>In Chap. 2 we have presented all the basic theory necessary to accomplish</p><p>meaningful probabilistic analyses of engineering problems. We have</p><p>defined, interpreted, and operated on the following important factors: sample</p><p>spaces, events, probabilities (marginal and conditional), random variables</p><p>(simple and joint), probability distributions (discrete, continuous, and mixed;</p><p>marginal, joint, and conditional), random variables which are functions of</p><p>random variables [Y = g(X)], expectation, and moments (marginal, joint, and</p><p>conditional).</p><p>These represent the fundamental tools for construction and analysis of</p><p>stochastic models. The next chapter will present a number of the most</p><p>frequently encountered models. It is emphasized throughout Chap. 3 that we</p><p>are merely applying there the methods of this chapter in conjunction with</p><p>various sets of assumptions. These assumptions represent the engineer’s</p><p>model of the physical phenomenon.</p><p>In Chap. 4 we discuss the problem of estimating the parameters of these</p><p>models when presented with a limited quantity of real data. We shall find</p><p>that again we are only applying the methods of Chap. 2 to a particular class</p><p>of random variables called statistics. Having analyzed the probabilistic</p><p>behavior (mean, variance, distribution) of a statistic, we attempt to quantify</p><p>the confidence we can place on inferences about the model that are drawn</p><p>from an observed value of the statistic.</p><p>Chapters 5 and 6 again apply the tools of this chapter to a particular class of</p><p>problems, namely, decision making in the face of uncertainty.</p><p>REFERENCES</p><p>General</p><p>Feller, W. [1957]: “An Introduction to Probability Theory and Its</p><p>Applications,” vol. I, 2d ed., John Wiley & Sons, Inc., New York.</p><p>Freeman, H. [1963]: “Introduction to Statistical Inference,” Addison-Wesley</p><p>Publishing Company, Inc., Reading,</p><p>Mass.</p><p>Hahn, G. J. and S. S. Shapiro [1967]: “Statistical Models in Engineering,”</p><p>John Wiley & Sons, Inc., New York.</p><p>Hald, A. [1952]: “Statistical Theory with Engineering Applications,” John</p><p>Wiley & Sons, Inc., New York.</p><p>Hammersley, J. M. and D. C. Handscomb [1964]: “Monte Carlo Methods,”</p><p>Methuen & Co., Ltd., London.</p><p>Parzen, E. [1960]: “Modern Probability Theory and Its Applications,” John</p><p>Wiley & Sons, Inc., New York.</p><p>Tocher, K. D. [1963]: “The Art of Simulation,” D. Van Nostrand Company,</p><p>Inc., Princeton, New Jersey.</p><p>Tribus, M. [1969]: “Rational Descriptions, Decisions, and Designs,”</p><p>Pergamon Press, New York.</p><p>Von Mises, R. [1957]: “Probability, Statistics, and Truth,” 2d ed., The</p><p>Macmillan Company, New York.</p><p>Wadsworth, G. P. and J. G. Bryan [I960]: “Introduction to Probability and</p><p>Random Variables,” McGraw-Hill Book Company, New York.</p><p>Specific text references</p><p>ACI Standard Recommended Practice for Evaluation of Compression Test</p><p>Results of Field Concrete (ACI Standard 214–65) [1965], American</p><p>Concrete Institute, Detroit, Michigan.</p><p>Benson, M. A. [1965]: Spurious Correlation in Hydraulics and Hydrology,</p><p>ASCE Proc, J. Hydraulics Div., vol. 91, no. HY4, July.</p><p>Blum, A. M. [1964]: Digital Simulation of Urban Traffic, IBM Systems J.,</p><p>vol. 3, no. 1, p. 41.</p><p>Esteva, L. and E. Rosenblueth [1964]: Espectros de Temblores a Distancias</p><p>Moderadas y Grandes, Bol. Soc. Mex. Ing. Sismica, vol. 2, no. 1, March.</p><p>Fishburn, P. C. [1964]: “Decision and Value Theory,” John Wiley & Sons,</p><p>Inc., New York.</p><p>Goldberg, J. E., J. L. Bogdanoff, and D. R. Sharpe [1964]: Response of</p><p>Simple Nonlinear Systems to a Random Disturbance of the Earthquake</p><p>Type, Bull Seismol. Soc. Am., vol. 54, no. 1, pp. 263–276, February.</p><p>Hognestad, E. [1951]: A Study of Combined Bending and Axial Load in</p><p>Reinforced Concrete Members, Eng. Exptl. Sta. Bull., no. 399, University of</p><p>Illinois.</p><p>Hufschmidt, M. and M. B. Fiering [1966]: “Simulation Techniques for</p><p>Design of Water Resource Systems,” Harvard University Press, Cambridge,</p><p>Miss.</p><p>Hutchinson, B. G. [1965]: The Evaluation of Pavement Structural</p><p>Performance, Ph. D. thesis, Department of Civil Engineering, University of</p><p>Waterloo, Waterloo, Ontario.</p><p>Rosenblueth, E. [1964]: Probabilistic Design to Resist Earthquakes, J. Eng.</p><p>Mech. Div., ASCE Proc, vol. 90, paper 4090, October.</p><p>PROBLEMS</p><p>2.1. An engineer is designing a large culvert to carry the runoff from two</p><p>separate areas. The quantity of water from area A may be 0, 10, 20, 30 cfs</p><p>and that from B may be 0, 20, 40, 60 cfs. Sketch the sample spaces for A and</p><p>B jointly and for A and B separately. Define the following events graphically</p><p>on the sketches.</p><p>images</p><p>2.2. A warehouse floor system is to be designed to support cartons filled</p><p>with canned food. The cartons are cubic in shape, 1 ft on a side and weigh</p><p>100 lb each. Consider that the cartons may be stacked to a height of 8 ft.</p><p>(a) Sketch the sample space for total weight on a square foot of floor area</p><p>assuming that it is loaded by one stack of boxes. How would this sample</p><p>space be changed if the area in question can be loaded by half the weight of</p><p>each of two stacks of boxes?</p><p>(b) Sketch the sample space for total load on two adjacent floor areas each 1</p><p>by 1 ft, assuming that each such area supports a single stack of boxes.</p><p>(c) Define the following events on the sketches.</p><p>images</p><p>2.3. (a) Sketch a sample space for the following experiment. The number of</p><p>vehicles on a bridge at a particular instant is going to be counted and</p><p>weighed; only the total number and total weight of the vehicles are to be</p><p>recorded (observed). The maximum number of vehicles which can be found</p><p>is five; the maximum weight of a single vehicle is 5 tons and the minimum</p><p>weight is 2 tons.</p><p>(b) Indicate on the sketch the regions corresponding to each of the following</p><p>events:</p><p>images</p><p>2.4. (a) Sketch a sample space for the following experiment: A timber pile</p><p>will be chosen from a supply of assorted lengths L, the longest of which is</p><p>60 ft. The pile will be driven into the ground in an area where the solid-rock-</p><p>bearing stratum is at a variable depth D, the maximum being 60 ft.</p><p>(b) On a sequence of such sketches shade the following events:</p><p>images</p><p>2.5. A question of the acceptability of an existing concrete culvert to carry</p><p>an anticipated flow has arisen. Records are sketchy, and the engineer assigns</p><p>estimates of annual maximum flow rates and their likelihoods of occurrence</p><p>(assuming that a maximum of 12 cfs is possible) as follows:</p><p>images</p><p>(a) Construct the sample space. Indicate events A, B, C, A ∩ C, A ∩ B, and</p><p>Ac ∩Bc on the sample space.</p><p>images</p><p>2.6. (a) An engineer has observed that two of the designers in his office</p><p>work at different rates and have a different frequency of making errors. If</p><p>designer A will take 6, 7, or 8 hr to do a particular job with the possibility of</p><p>0, 1, 2 errors, sketch the sample space associated with this designer.</p><p>Designer B is faster but more prone to errors. He will require 5, 6, or 7 hr</p><p>and may make 0, 1, 2, 3 errors. Sketch the sample space for designer B.</p><p>If designer A is equally likely to have any one of the nine possible</p><p>combinations occur, sketch the probability attributes on the sample space for</p><p>A.</p><p>If designer B is twice as likely to make 1 error as either 0, 2, or 3 errors and</p><p>twice as likely to require 6 hr as either 5 or 7 hr, sketch the probability</p><p>attributes on the sample space for designer B. Assume independence.</p><p>(b) Sketch following events for designer A and determine probabilities.</p><p>images</p><p>(c) Indicate the following events for designer B on sample space and</p><p>determine probabilities.</p><p>images</p><p>(d) Compare the probabilities of various job times for the two designers</p><p>(sketch).</p><p>(e) If both designers are working at the same time on the same kind of task,</p><p>can the probabilities of two 7-hr times be added to determine the probability</p><p>of a 14-hr total? Why?</p><p>(f) Determine the probabilities of various total time requirements in (e) if the</p><p>designers work independently. Repeat for total number of errors.</p><p>2.7. If the occurrences of earthquakes and high winds are unrelated, and if,</p><p>at a particular location, the probability of a “high” wind occurring</p><p>throughout any single minute is 10-5 and the probability of a “moderate”</p><p>earthquake during any single minute is 10–8:</p><p>(a) Find the probability of the joint occurrence of the two events during any</p><p>minute. Building codes do not require the engineer to design the building for</p><p>the combined effects of these loads. Is this reasonable?</p><p>(b) Find the probability of the occurrence of one or the other or both during</p><p>any minute. For rare events, i.e., events with small probabilities of</p><p>occurrence, the engineer frequently assumes</p><p>images</p><p>Comment.</p><p>(c) If the events in succeeding minutes are mutually independent, what is the</p><p>probability that there will be no moderate earthquakes in a year near this</p><p>location? In 10 years? Approximate answers are acceptable.</p><p>2.8. Revise the water-supply situation (page 52) for a lateral with four</p><p>tributary sites. Determine probabilities of various water demand levels.</p><p>What is the optimum design size if capacity costs are linear at $500 per unit</p><p>provided and enlarging costs are $1000 per unit provided? Formulate a</p><p>solution for arbitrary costs if enlarging costs are double initial capacity costs</p><p>and both are linear with quantity.</p><p>2.9. Find the probability that a pile will reach bedrock n ft below without</p><p>hitting a rock if the probability is p that such a rock will be struck in any foot</p><p>and the occurrences of rocks in different 1-ft levels are mutually</p><p>independent events.</p><p>Evaluate this probability for images and n = 10 and 20 feet, and for</p><p>images and n = 10 and 20 ft. Notice that even in the last case it is not</p><p>certain that a rock will be struck.</p><p>2.10. Consider the possible failure of a water supply system to meet the</p><p>demand during any given summer day.</p><p>(a) Use the equation of total probability [Eq. (2.1.12)] to determine the</p><p>probability that the supply will be insufficient if the probabilities</p><p>shown in</p><p>the table are known.</p><p>images</p><p>(b) Find the probability that a demand level of 150,000 gal/day was the</p><p>“cause” of the system’s failure to meet demand if an inadequate supply was</p><p>observed. (Clearly the word “cause” is not appropriate in such a situation,</p><p>but this interpretation of Bayes theorem is often adopted. It should be used</p><p>with caution.)</p><p>(c) The likelihood of a pump failing and causing the system to fail is 0.02</p><p>regardless of the demand level. What does the equation of total probabilities</p><p>reduce to in this case, that of independence?</p><p>(d) The system may fail in one and only one of three possible modes: M1,</p><p>inadequate supply; M2, a pump failure; or M3, overload of the purification</p><p>plant. We have the following information:</p><p>images</p><p>(The other probabilities are as given above.) Find the probabilities of each of</p><p>the various possible causes (modes) if a system failure takes place when the</p><p>demand level is 150,000 gal/day. Hint: Modify Bayes theorem [Eq. (2.1.13)]</p><p>to read P[A | Bt n</p><p>images</p><p>in which here the Bi’s are the modes of failure, the Ci is the demand level,</p><p>and A is the event a failure took place. Verify this “conditional” Bayes</p><p>theorem. It can be interpreted as a Bayes-theorem application in the</p><p>conditional sample space (i.e., given Ci).</p><p>(e) What are the probabilities of the various causes (modes) if the demand</p><p>level at failure was 100,000 gal/day? In general, what does Bayes theorem</p><p>reduce to if A can occur if and only if Bi occurs? Recognize that within the</p><p>familiar deterministic, one-cause-one-effect view of phenomena, the</p><p>determination of cause by observation of effect can be interpreted as just</p><p>such a special case of the application of Bayes theorem.</p><p>2.11. An engineer concerned with providing a continuous water supply to a</p><p>critical operation is considering installing a second “backup” pump to take</p><p>the place of the primary pump in the event of its failure. Let F1 be the event</p><p>that the primary pump fails once during a given period. (The likelihood of</p><p>two or more such failures is negligible). Let F2 be the event that the second</p><p>pump fails to function if it is switched on.</p><p>(a) What is the relationship between these events and the event F0 that the</p><p>system fails to provide continuous service during the period?</p><p>(b) What is the reliability of the system images in terms of the reliabilities</p><p>of the individual pumps images if the events F1 and F2 are independent?</p><p>How does this answer compare with the reliability of a series-type system in</p><p>which both components must operate simultaneously for the system to</p><p>function?</p><p>(c) What is P[F0] if P[F1] = P[F2] = 10–1 and the conditions in (b) hold?</p><p>(d) If the failure of one component in a redundant system is caused by an</p><p>overload, the failure of the stand-by unit will probably not be independent of</p><p>the failure of the first. Generally, P[F2 | F1]> P[F2]. Find the reliability of</p><p>the system above if P[F2| F1]= 2 X 10–1.</p><p>The system described here is an example of a parallel-type system, in which</p><p>redundant elements are provided to reduce the likelihood of a system failure.</p><p>2.12. In the study of a storage-dam design, it is assumed that quantities can</p><p>be measured sufficiently accurately in units of ¼ of the dam’s capacity. It is</p><p>known from past studies that at the beginning of the first (fiscal) year the</p><p>dam will be either full, ¾ full, ½ full, or ¼ full, with probabilities ⅓, ⅓,</p><p>images, and images, respectively. During each year water is released.</p><p>The amount released is ½ the capacity if at least this much is available; it is</p><p>all that remains if this is less than ½ the capacity. After release, the inflow</p><p>from the surrounding watershed is obtained. It is either ½ or ¼ of the dam’s</p><p>capacity with probabilities ⅔ and ⅓, respectively. Inflow causing a total in</p><p>excess of the capacity is spilled. Assuming independence of annual inflows,</p><p>what is the probability distribution of the total amount of water at the</p><p>beginning of the third year?</p><p>2.13. A large dam is being planned, and the engineer is interested in the</p><p>source of fine aggregate for the concrete. A likely source near the site is</p><p>rather difficult to survey accurately. From surface indications and a single</p><p>test pit, the engineer believes that the magnitude of the source has the</p><p>possible descriptions: 50 percent of adequate; adequate; or 150 percent of</p><p>possible demand. He assigns the following probabilities of these states.</p><p>images</p><p>Prior to ordering a second test pit, the engineer decides that the various</p><p>likelihoods of the sample’s possible indications (Z1 Z2, Z3) depend upon the</p><p>(unknown) true state as follows:</p><p>images</p><p>What are the probabilities of observing the various events Z1, Z2, and Z3?</p><p>The second test pit is dug and the source appears adequate from this pit.</p><p>Compute the posterior probabilities of state. If another test pit gives the</p><p>same result, calculate the second set of posterior state probabilities.</p><p>Compare prior and posterior state probabilities.</p><p>2.14. The following “urn model” has been proposed to model the occurrence</p><p>(black ball) or nonoccurrence (red ball) of rainfall on successive days. There</p><p>are three urns: the “initial urn” containing n1 black and m1 red balls, the “dry</p><p>urn” containing n2 black and m2 red balls, and the “rainy urn” containing n3</p><p>black and m3 red balls. To simulate (sample) a sequence, one draws a ball</p><p>from the initial urn, its color indicating occurrence or not of rain on the first</p><p>day. The weather on the second day is found by sampling from the rainy or</p><p>dry urn depending on the outcome of the first trial. Subsequent draws are</p><p>made from the rainy or dry urn depending on the weather on the immediate</p><p>past day. The ball is returned to its appropriate urn after each draw. The</p><p>model is devised to simulate the persistence of rainy or dry spells. This is an</p><p>example of what we will come to know as a Markov chain (Sec. 3.7).</p><p>Assume that all the balls in any urn are equally likely to be drawn and n1 =</p><p>n2 = n3 = 10 and m1 = 70, m2 = 90, and m3 = 40.</p><p>(a) What is the probability that the first four days in a row will be observed</p><p>to be rainy?</p><p>(b) What is the probability that at least three dry days will follow a rainy</p><p>day? Source: E. H. Wiser [1965], Modified Markov Probability Models of</p><p>Sequences of Precipitation Events, Monthly Weather Review, vol. 93, pp.</p><p>511–516. This reference contains many suggested urn models of more</p><p>complicated varieties.</p><p>2.15. At a traffic signal, the number N of cars that arrive during the red-</p><p>green cycle on the northbound leg has a PMF of px(n), n = 0, 1, 2, ... . At</p><p>most three cars can pass through the intersection in a cycle. The engineer is</p><p>disturbed by his choice of cycle times any time there is a car left at the end</p><p>of the green phase.</p><p>(a) What is the probability that the engineer is disturbed with any particular</p><p>cycle if at the end of the previous green phase no cars are present?</p><p>(b) What is the probability that he is “disturbed” at least twice in succession?</p><p>(Assume the same zero starting conditions as in (a) at the beginning of the</p><p>first of these two cycles.)</p><p>(c) Evaluate (a) and (b) if</p><p>(See Table A.2.)</p><p>2.16. A quality-control plan for the concrete in a nuclear reactor containment</p><p>vessel calls for casting 6 cylinders for each batch of 10 yd3 poured and</p><p>testing them as follows:</p><p>1 at 7 days</p><p>1 at 14 days</p><p>2 at 28 days</p><p>2 more at 28 days if any of first four cylinders is “inadequate”</p><p>The required strength is a function of age.</p><p>If the cylinder to be tested is chosen at random from those remaining (i. e.,</p><p>with equal likelihoods):</p><p>(a) What is the probability that all six will be tested if in fact one inadequate</p><p>cylinder exists in the six?</p><p>(b) If the batch will be “rejected” if two or more inadequate cylinders are</p><p>found, what is the likelihood that it will not be rejected given that exactly</p><p>two are in fact inadequate? (Rejection will lead to more expensive coring</p><p>and testing of concrete in place.)</p><p>(c). A “satisfactory” concrete batch gives rise to an inadequate cylinder with</p><p>probability p = 0.1. (This value is consistent</p><p>with present recommended</p><p>practice.) What is the probability that there will be one or more inadequate</p><p>cylinders in the six when the batch is “satisfactory”? (Assume independence</p><p>of the quality of the individual cylinders.)</p><p>(d) Given that the batch is satisfactory (p = 0.1), what is the probability that</p><p>the batch will be rejected? What is the probability that an unsatisfactory</p><p>batch (in particular, say, p = 0.3) will not be rejected? Clearly a quality</p><p>control plan wants to keep both these probabilities low, while also keeping</p><p>the cost of testing small.</p><p>2.17 new transportation system has three kinds of vehicles with seating</p><p>capacities 2, 4, and 8. They become available to the dispatcher at a terminal</p><p>in mixed trains having from one to four cars. If each of the possible train</p><p>lengths is equally likely and if the three vehicles appear independently and</p><p>in equal relative frequencies, what is the probability that exactly 10 seats</p><p>will be available for dispatch in an arbitrary train?</p><p>2.18. Pairwise independence does not imply mutual independence.</p><p>Assume that the particular scales used in dry-batching a concrete mix are</p><p>such that both aggregate and cement weights are subject to error. The two</p><p>weights are measured independently. The aggregate is equally likely to be</p><p>measured exactly correct or 20 pounds too large. The cement is equally</p><p>likely to be measured exactly correct or 20 pounds too small. The total</p><p>weight of aggregate and cement is desired. Let event A be no error in</p><p>aggregate weight measurement, event B be no error in cement weight, and</p><p>event C be no error in total weight.</p><p>(a) Show that the pairs of events A and B, A and C, and B and C are</p><p>independent.</p><p>(b) Use (a) as a counterexample to show that pairwise independence does</p><p>not imply mutual independence.</p><p>(c) Is the inverse statement true?</p><p>2.19. A major city transports water from its storage reservoir to the city via</p><p>three large tunnels. During an arbitrary summer week there is a probability q</p><p>that the reservoir level will be low. Owing to the occasional call to repair a</p><p>tunnel or its control valves, etc., there are probabilities pi(i = 1, 2, 3) that</p><p>tunnel i will be out of service during any particular week. These calls to</p><p>repair particular tunnels are independent of each other and of the reservoir</p><p>level.</p><p>The “safety performance” of the system (in terms of its potential ability to</p><p>meet heavy emergency fire demands) in any week will be satisfactory if the</p><p>reservoir level is high and if all tunnels are functioning; the performance</p><p>will be poor if more than one tunnel is out of service or if the reservoir is</p><p>low and any tunnel is out of service; the performance will be marginal</p><p>otherwise.</p><p>(a) Define the events of interest. In particular, what events are associated</p><p>with marginal performance?</p><p>(b) What is the probability that exactly one tunnel fails?</p><p>(c) What is the probability of marginal performance?</p><p>(d) What is the probability that any particular week of marginal performance</p><p>will be caused by a low reservoir level rather than by a tunnel being out of</p><p>service?</p><p>2.20. At a certain intersection, of all cars traveling north, the relative</p><p>frequency of cars continuing in the same direction is p. The relative</p><p>frequency of those turning east is q; all others turn west.</p><p>Assume that drivers behave independently of one another. A small group of</p><p>n cars enters the intersection. For this group</p><p>(a) What is the marginal distribution of Y, the number of cars turning west?</p><p>Find the conditional distribution of X, the number of cars turning east, given</p><p>that Y equals y. Hint: what is the probability that any car not turning west</p><p>will turn east?</p><p>(b) Find the joint distribution of X and Y. Be careful with the limits of</p><p>validity.</p><p>(X, Y, and Z, the number going straight, have a joint “multinomial”</p><p>distribution which is studied in Sec. 3.6.1.)</p><p>2.21. Two kinds of failure of reinforced-concrete beams concern the</p><p>engineer: one, “the under-reinforced moment” failure, is preceded by large</p><p>deflections which give warning of its imminence; the other, the “diagonal-</p><p>tension or shear” failure, occurs suddenly and without warning, not</p><p>permitting persons to remove the cause of the overload or to evacuate the</p><p>structure.</p><p>A structural consultant has been retained to observe a suspect beam in a</p><p>building. From the engineer’s experience he estimates that about 5 percent</p><p>of all beams proportioned according to the building code in use at the time</p><p>the building was designed will fail owing to a weakness in the shear manner,</p><p>if tested to failure, while the others will fail in the moment manner. From</p><p>laboratory experience, however, the engineer knows that at some load prior</p><p>to failure 8 of 10 beams destined to fail in the shear manner will exhibit</p><p>small characteristic diagonal cracks near their ends. On the other hand, only</p><p>1 of 10 beams which would finally fail in the moment manner shows similar</p><p>cracks prior to failure.</p><p>Suppose that the relative consequences of the sudden shear failure versus</p><p>warning-giving moment type of failure are such that the expensive</p><p>replacement of the beam is justified only if a sudden failure is more likely</p><p>than a moment failure. Then, if upon inspecting the beam, the engineer</p><p>observes these characteristic diagonal cracks, should he demand the repairs</p><p>or conclude that the risk is too small to justify the repairs (without further</p><p>study)?</p><p>2.22. A preliminary investigation of a site leads an engineer to state that the</p><p>relative weights are 3 to 5 to 2 (respectively) that the unconfined</p><p>compressive strength of the soil below is 1200, 1000, or 800 psf (the only</p><p>three values considered possible, for simplicity). “Undisturbed” samples of</p><p>the soil will be obtained by boring and tested to gain further information.</p><p>Owing to the difficulties in obtaining such samples and owing to testing</p><p>inaccuracies, the following frequencies of indicated strengths are considered</p><p>applicable for each specimen:</p><p>P [indicated strength | state]</p><p>images</p><p>The engineer calls for a sampling plan of two independent specimens.</p><p>(a) Find the conditional probabilities of each of the possible outcomes of</p><p>this sample of size two given that the true strength is 1200 psf.</p><p>(b) If the results of the sampling were one specimen indicating 1000 and one</p><p>indicating 800 psf, find the engineer’s posterior probabilities of the strength.</p><p>(c) Suppose that after these two specimens the engineer continued sampling</p><p>and found an uninterrupted sequence of specimens indicating 1200 psf.</p><p>After how many could he stop:</p><p>(i) Confident that the strength was not actually 800?</p><p>(ii) At least “90 percent confident” that the strength was actually 1200?</p><p>2.23. Consider the following problem associated with synchronizing traffic</p><p>lights. A particular traffic light has a cycle as follows:</p><p>Red = 1 min</p><p>Green = 1.5 min</p><p>Yellow =0.5 min</p><p>Some distance before this light—light 1—is another light—light 2. Owing</p><p>to varying drivers and conditions, the travel time between the two varies</p><p>from vehicle to vehicle. Data suggest that 40 percent of all cars leaving the</p><p>location of light 2 at a time when light 1 is red are not delayed by light 1</p><p>when they reach it, 80 percent of all cars leaving light 2 during a green cycle</p><p>of light 1 are not delayed, and 20 percent of all cars leaving during a yellow</p><p>cycle are not delayed.</p><p>Given that a car was delayed by light 1, what is the probability that it left</p><p>light 2 while light 1 was red? green? yellow?</p><p>2.24. A machine to detect improper welds in a fabricating shop detects 80</p><p>percent of all improper welds, but it also incorrectly indicates an improper</p><p>weld on 5 percent of all satisfactory welds. Past experience indicates that 10</p><p>percent of all welds are improper. What is the probability that a weld which</p><p>the machine indicates to be defective is in fact satisfactory?</p><p>2.25. The cost of running an engineering office is a function of office size X.</p><p>Assume that an engineer is trying to make a projection of cost for the next</p><p>year’s operations. He believes the demand will require an X of from</p><p>1 to 6.</p><p>images</p><p>(a) Cost varies with X according to:</p><p>images</p><p>The first term represents space and salary expense while the second term</p><p>represents overhead. Find the PMF of Y.</p><p>(b) The gross income to the owner Z is jointly distributed with X:</p><p>images</p><p>Find the PMF of the net income:</p><p>images</p><p>2.26. Show that the function below is the PDF of R, the distance between the</p><p>epicenter of an earthquake and the site of a dam, when the epicenter is</p><p>equally likely to be at any location along a neighboring fault. You may</p><p>restrict your attention to a length of the fault l that is within a distance r0 of</p><p>the site because earthquakes at greater distances will have negligible effect</p><p>at the site.</p><p>images</p><p>Sketch this function.</p><p>images</p><p>Fig. P2.26</p><p>2.27. A system has a certain capacity R and must meet a maximum demand</p><p>L; both can be treated as nonnegative, continuous random variables, with</p><p>joint density fR,L(r,l).</p><p>Write an integral expression in terms of f R,L(r,l) for the probability of</p><p>failure, PF = P[R</p><p>contains two</p><p>histograms of the data of Table 1.1.1, illustrating the influence of interval</p><p>size. An empirical practical guide has been suggested by Sturges [1926]. If</p><p>the number of data values is n, the number of intervals k between the</p><p>minimum and maximum value observed should be about</p><p>images</p><p>Fig. 1.1.2 Cumulative frequency distribution of floor-load data.</p><p>images</p><p>in which logarithms to the base 10 should be employed. Unfortunately, if</p><p>the number of values is small, the choice of the precise point at which the</p><p>interval divisions are to occur also may alter significantly the appearance of</p><p>the histogram. Examples can be found in Sec. 1.2 and in the problems at</p><p>the end of this chapter. Such variations in shape may at first be</p><p>disconcerting, but they are indicative of a failure of the set of data to</p><p>display any sharply defined features, a piece of information which is in</p><p>itself valuable to the engineer. This failure may be because of the</p><p>inadequate size of the data set or because of the nature of the phenomenon</p><p>being observed.</p><p>images</p><p>Fig. 1.1.3 Influence of interval size on appearance of histogram (data of</p><p>Table 1.1.1).</p><p>1.2 NUMERICAL SUMMARIES</p><p>Central value measures The single most helpful number associated with a</p><p>set of data is its average value, or arithmetic mean. If the sequence of</p><p>observed values is denoted x1, x2, . . ., xn, the sample mean images is</p><p>simply</p><p>images</p><p>Fifteen reinforced-concrete beams built by engineering students to the same</p><p>specifications and fabricated from the same batch of concrete were tested in</p><p>flexure. The observed results of first-crack and ultimate loads, recorded to</p><p>the nearest 50 lb, are presented in Table 1.2.1. Their cumulative frequency</p><p>distributions are shown in Fig. 1.2.1. (They were plotted by the method</p><p>suggested in the footnote on page 7.)† The scatter of the data might be</p><p>attributed to unrecorded construction and testing differences, inconsistent</p><p>workmanship, human errors, and inherent material variability as well as</p><p>observation and measurement errors. The mean value of the failure loads is</p><p>computed to be</p><p>Table 1.2.1 Tests of identical reinforced concrete beams</p><p>images</p><p>images</p><p>Fig. 1.2.1 Cumulative frequency distributions; beam data.</p><p>images</p><p>The sample mean is frequently interpreted as a typical value or a central</p><p>value of the data. If required to give only a single number, one would</p><p>probably use this sample mean as his “best prediction” of the load at which</p><p>a nominally identical beam would fail.</p><p>Other measures of the central tendency of a data set include the mode, the</p><p>most frequently occurring value in the data set, and the median, the middle</p><p>value in an ordered list (the middle value if n is odd, or the average of the</p><p>two middle values if n is even) (see Table 1.2.2). The order of observing the</p><p>values is usually not important, and they may be arranged in any way</p><p>which is convenient. The median of the failure data is 9900 lb; the mode is</p><p>not unique, since several values appear twice and none appears more times.</p><p>These two terms occur commonly in the literature of other fields and</p><p>consequently should be understood, but they only infrequently prove</p><p>meaningful in engineering problems.</p><p>Table 1.2.2 Ordered first-crack and ordered failure-load data</p><p>images</p><p>Measures of dispersion Given a set of observed data, it is also desirable to</p><p>be able to summarize in a single number something of the variability of the</p><p>observations. In the past the measure most frequently occurring in</p><p>engineering reports was the range of the data. This number, which is</p><p>simply the difference between the maximum and minimum values</p><p>observed, has the virtue of being easy to calculate, but certain obvious</p><p>weaknesses have led to its being replaced or supplemented. The range</p><p>places too much emphasis on the extremes, which are often suspected of</p><p>experimental error, and neglects the bulk of the data and experimental</p><p>effort which lies between these extremes. The range is also sensitive to the</p><p>size of the sample observed, as will be demonstrated in Sec. 3.3.3.</p><p>A far more satisfactory measure of dispersion is found in the sample</p><p>variance. It is analogous to the moment of inertia in that it deals with</p><p>squares of distances from a center of gravity, which is simply the sample</p><p>mean. The sample variance s2 is defined here to be</p><p>images</p><p>To eliminate the dependence on sample size, the squared distances are</p><p>divided by n to yield an average squared deviation. There are sound reasons</p><p>favoring division by n – 1, as will be shown in Chap. 4, but the intuitively</p><p>more satisfactory form will not be abandoned until the reader can</p><p>appreciate the reasons. Similarly, small computational changes in other</p><p>definitions given in this chapter may be found desirable in the light of later</p><p>discussions on the estimation of moments of random variables (Secs. 2.4</p><p>and 4.1).</p><p>Expansion of Eq. (1.2.2a) yields an expression which will be found far</p><p>more convenient for computation of s2:</p><p>images</p><p>But, by Eq. (1.2.1),</p><p>images</p><p>Therefore,</p><p>images</p><p>The positive square root s of the sample variance of the data is termed the</p><p>sample standard deviation. It is analogous to the radius of gyration of a</p><p>structural cross section; they are both shape- rather than size-dependent</p><p>parameters. The addition of a constant to all observed values, for example,</p><p>would alter the sample mean but leave the sample standard deviation</p><p>unchanged. This number has the same units as the original data, and, next</p><p>to the mean, it conveys more useful information to the engineer than any</p><p>other single number which can be computed from the set of data. Roughly</p><p>speaking, the smaller the standard deviation of the sample, the more</p><p>clustered about the sample mean is the data and the less frequent are large</p><p>variations from the average value.</p><p>For the beam-failure data the sample variance and sample standard</p><p>deviation are computed as follows:</p><p>images</p><p>(Notice that, owing to the required subtraction of two nearly equal</p><p>numbers, many significant digits must be carried in the sums to maintain</p><p>accuracy.) In this example, the sample standard deviation might be used to</p><p>compare the variability of the strength of lab beams with that of field-</p><p>constructed beams.</p><p>When comparing the relative dispersion of more than one kind of data, it is</p><p>convenient to have a dimensionless description such as the commonly</p><p>quoted sample coefficient of variation. This quantity υ is defined as the</p><p>ratio of the sample standard deviation to the sample mean.</p><p>images</p><p>The sample coefficient of variation of the beam-failure data is</p><p>images</p><p>while that of the first observed crack is much larger, being</p><p>images</p><p>The engineer might interpret the difference in magnitudes of these</p><p>coefficients as an indication that first-crack loads are “more variable” or</p><p>more difficult to predict closely than failure loads. Such information is</p><p>important when appearance as well as strength is a design criterion.</p><p>Measure of asymmetry One other numerical summary† of observed data</p><p>is simply a logical extension of the reasoning leading to the formula for the</p><p>sample variance. Where the variance was an average second moment about</p><p>the mean, so the sample coefficient of skewness is related to the third</p><p>moment about the mean. To make the coefficient nondimensional, the</p><p>moment is divided by the cube of the sample standard deviation.</p><p>The coefficient of skewness g1 provides a measure of the degree of</p><p>asymmetry about the mean of the data:</p><p>images</p><p>The coefficient is positive for histograms skewed to the right (i.e., with</p><p>longer tails to the right) and negative for those skewed to the left. Zero</p><p>skewness results from symmetry but does not necessarily imply it.</p><p>For the beam-failure data of Table 1.2.1,</p><p>images</p><p>indicating mild skewness to the right. The implication is that there were</p><p>fewer but larger deviations to the high side than to the low side of the</p><p>average value. (The sample coefficient of skewness should be calculated</p><p>using images since an expansion similar to Eq. (1.2.2b) for the sample</p><p>variance does not prove useful.)</p><p>load stress rX, where r is the</p><p>modulus of elasticity of the material:</p><p>images</p><p>(a) Find fY(y).</p><p>(b) Find E[Y] two ways, using both fY(y) and fX(x).</p><p>2.36. (a) The Bernoulli distribution:</p><p>images</p><p>for 0 ≤ p ≤ 1. Find the mean and variance of X in terms p. For what value of</p><p>p is the variance a maximum? Evaluate at p = 0.5 and 0.1.</p><p>(b) The Poisson distribution:</p><p>images</p><p>for λ > 0. Find the mean and standard deviation of X. Evaluate at λ = 0.5 and</p><p>2. Sketch the PMF for these cases.</p><p>(c) A triangular distribution:</p><p>images</p><p>for a and b nonnegative. Find mean and standard deviation of X. It will</p><p>prove simplest to make use of tables of moments of areas of simple shapes</p><p>and parallel axes transformation theorems.</p><p>2.37. Owing to the gradual accumulation of strain, the likelihood (given that</p><p>the last earthquake took place in year zero) of a major earthquake on a</p><p>particular fault in year i grows with i. Specifically, it can perhaps be</p><p>assumed that</p><p>images</p><p>in which a is a constant between 0 and 1.</p><p>(a). What is the probability that the first occurrence will take place in year</p><p>k?</p><p>(b). What is the cumulative distribution function of X, the year of the first</p><p>occurrence?</p><p>2.38. Population concentration in cities has been found to obey the law</p><p>images</p><p>as a function of radius r from the center. Convert the law to a PDF and find</p><p>the outermost radius necessary in a public transportation network that will</p><p>serve 75 percent of the residents.</p><p>Find the average distance from the center of the city of</p><p>(a) A resident</p><p>(b) A resident served by the network above</p><p>2.39. In a construction project, units are arriving at an operation which takes</p><p>time or causes delay. If a unit arrives within b sec of the arrival of the</p><p>previous unit, departure of the later unit will be delayed until a given time c</p><p>sec after the arrival of the</p><p>images</p><p>Fig. P2.39</p><p>preceding object (c > b). If a unit arrives more than b sec after the previous</p><p>arrival, it will be delayed until d sec (d</p><p>the stress</p><p>distribution on the concrete portion of the section. All might be treated as</p><p>independent random variables.</p><p>Find the approximate mean and variance of M given:</p><p>images</p><p>Which variables “contribute” most significantly to the variance of M? What</p><p>are the implications to the engineer seeking ecomonical ways to reduce the</p><p>variance of M ?</p><p>2.47. Standardized variables, U = (x — mX)/σX-’</p><p>(a) Show that</p><p>images</p><p>For these reasons the variable U = (X — mx)σx is called the unit,</p><p>standardized, or normalized variable. Recall that the distribution of U is of</p><p>exactly the same shape as that of A”. Consequently, it is commonly used to</p><p>facilitate tabulating distribution functions (see Sec. 3.3.1).</p><p>(b) Show that the correlation coefficient between any two random variables</p><p>A” and Z is the covariance between their corresponding standardized</p><p>variables</p><p>images</p><p>2.48. In the determination of the strength of wooden structural members it</p><p>has been suggested that the estimated strength of full-size pieces A” can be</p><p>found as the product of the clear-wood strength of small, standard specimens</p><p>Y, times a strength ratio R. After inspection of data, Y and R have been</p><p>modeled as independent (normal, see Sec. 3.3.1) random variables. Find the</p><p>mean and variance of A in terms of the corresponding moments of R and Y</p><p>(see Prob. 2.52).</p><p>For construction grade Douglas Fir in bending, mR = 0.659, σR = 0.165, my</p><p>= 7480 psi, and VY = 15 percent. Evaluate m and σx. How many standard</p><p>deviations below the mean is the present allowable working stress of 1500</p><p>psi?</p><p>2.49. Uncorrelated variables through a linear transformation</p><p>(orthogonalization). Consider two random variables X and Y, not</p><p>necessarily independent.</p><p>(a) Find the covariance of Z and W when</p><p>images</p><p>in terms of the moments of A and Y.</p><p>(b) Find the values of the coefficients a, b, c, and d such that Z and W will be</p><p>uncorrelated.</p><p>(c) If X and Y are the monthly rainfalls at two neighboring locations, then Z</p><p>and W, with the coefficient properly chosen, can be considered the</p><p>uncorrelated rainfalls at two “fictitious” locations. When faced with the</p><p>problem of creating artificial rainfall records for water-resource systems,</p><p>Hufschmidt and Fiering [1966] have suggested that the (spatially)</p><p>uncorrelated records Z and W be generated as independent variables† (a</p><p>relatively easy task) and that these be properly combined to obtain values X</p><p>and Y for the actual (correlated) stations. What should the constants be in the</p><p>equations for X and Y,</p><p>images</p><p>(in terms of the moments of X and Y)? Hint: find a', b', c', and d' in terms of</p><p>a, b, c, and d first, by solving the pair of simulateous equations.</p><p>(d) Generate, using random numbers and the method above, a year’s record</p><p>of monthly rainfalls of two stations, each with (for simplicity) independent</p><p>(in time) monthly flows which have constant means m = 1 and standard</p><p>deviations a = 0.25. Assume that owing to the stations’ proximity to one</p><p>another, each pair of monthly rainfalls has a correlation coefficient of 0.7.</p><p>Assume normal distributions.</p><p>Note: The technique is obviously easily extended to n stations. The</p><p>procedure here is analogous to finding normal modes of vibration.</p><p>2.50. The cost of an operation is proportional to the square of the total time</p><p>required to complete it. Completion time for the first phase is X and the</p><p>second is Y. X and Y are correlated random variables with moments mX, mY,</p><p>σX, σY, and ρ X,Y.</p><p>Find</p><p>images</p><p>2.51. Linear transformation. Sketch an arbitrary density function fX(x) and</p><p>then sketch density functions for Y = a + bX for the special cases</p><p>images</p><p>The first three cases are particularly important; they represent the effect on</p><p>changing scale or units. Cases (d) and (e) illustrate the effect of the addition</p><p>of constants. Describe in words the effects on the PDF of changing units or</p><p>adding constants.</p><p>2.52. Distribution and moments of a product of two random variables.</p><p>The expression relating one random variable Z to two other random</p><p>variables:</p><p>images</p><p>is very common in engineering. For example, let X be the demand and Y be</p><p>the cost per unit of demand.</p><p>(a) If X and Y are independent continuous random variables, show that</p><p>images</p><p>and find fz(z) if</p><p>images</p><p>(b) Show that the variance of the product of independent random variables is</p><p>images</p><p>Hint: Use Eq. (2.4.25): Var [XY]= E[X2Y2]– [E[XY]]2.</p><p>(c) Evaluate the variance of Z for the PDF’s given above in two ways, using</p><p>the equation in b and using the PDF of Z computed in a.</p><p>2.53. The following technique (algorithm) can be used to generate a</p><p>sequence† of (pseudo) random numbers for simulation studies or other</p><p>purposes.</p><p>(a) Pick arbitrarily an initial number M0, which is less than 9999 and not</p><p>divisible by 2 or 5.</p><p>(b) Choose a constant multiplier M of the form</p><p>images</p><p>where t is any integer and r is any of the values 3, 11, 13, 19, 21, 29, 37, 53,</p><p>59, 61, 67, 69, 77, 83, 91. (A good choice of M is a value around 100.)</p><p>(c) Compute the product M1= MM0. The last four digits form the first ‘</p><p>‘random” number.</p><p>(d) Determine successive random numbers by forming the product Mi+1 =</p><p>MMi and retaining the last four digits only.</p><p>(i) Generate by hand a sequence of five random numbers and use the</p><p>numbers to simulate a sequence of tosses of an unbalanced coin with the</p><p>probability of “heads” equal to 0.7.</p><p>(ii) Write a computer subroutine to generate such random numbers.</p><p>2.54. The following relationships arise in the study of earthquake-resistant</p><p>design, where Y is ground-motion intensity at the building site, X is the</p><p>magnitude of an earthquake, and c is related to the distance between the site</p><p>and center of the earthquake:</p><p>images</p><p>If X is exponentially distributed (Sec. 3.2.2),</p><p>images</p><p>Show that the cumulative distribution function of Y, FY(y), is</p><p>images</p><p>Sketch this distribution.</p><p>2.55. Specifications are set so that all but a fraction p of the tested specimens</p><p>will be satisfactory. The number of specimens that have to be inspected</p><p>before an unsatisfactory one is found is X. The additional number before the</p><p>second is found is Y. Assume that X and Y are independent and</p><p>geometrically distributed (as will be shown in Sec. 3.1.3):</p><p>images</p><p>Find the probability mass function of Z = X + Y. What is Z in words?</p><p>2.56. The amount of water lost from a dam due to evaporation during a</p><p>summer is proportional to the dam’s surface area. The proportionality factor</p><p>is random, depending as it does on weather conditions. For a particular</p><p>reservoir with a particular cross section the surface area increases</p><p>proportionately to the volume of water in storage. The water in storage</p><p>during the summer (July and August) is the difference between the water</p><p>made available since the previous September and the water released since</p><p>that time. Both these volumes are random. Assume that the water released</p><p>and the water made available during the summer are negligible. Neglecting</p><p>second-order effects (e.g., the change in surface area due to the evaporation</p><p>loss), find the mean and variance of the water lost due to evaporation in a</p><p>summer in terms of the same moments of the various factors mentioned.</p><p>Assume a lack of correlation among these factors. Define carefully all the</p><p>variables, constants, moments, relationships, etc., that you use.</p><p>2.57. Reconsider the dam-operating-policy problem (Prob. 2.32).</p><p>(a) Find the expected value of Z, the water available.</p><p>(b Find the expected value of Y = g(Z), the water released.</p><p>2.58. Variability in stream flows makes it difficult to properly design the</p><p>capacity of new dams. Concern is primarily with providing sufficient storage</p><p>to avoid water shortage. Early studies focused on the range R, the difference</p><p>between the maximum value of the impounded water and its minimum value</p><p>over a period of n years, the design life of the dam. Assuming</p><p>(unrealistically, but as a first approximation) that the annual</p><p>“impoundments” are independent with common mean m and variance σ2, it</p><p>has been shown† that (for large n)</p><p>images</p><p>(a) Show that the</p><p>coefficient of variation of R is a constant, independent of</p><p>the stream characteristics m and σ and of the lifetime n.</p><p>(b) A possible dam design rule is to design for a range of value r0 = mR +</p><p>kσR, where k depends on the risk level adopted. For k = 2 how many times</p><p>the mean range is the design value? How many of (the infinite number of)</p><p>moments of the annual impoundment does the design range depend on? Is</p><p>design for an infinite lifetime possible? How sensitive to the design lifetime</p><p>is the design range? (For example, if n is doubled from 50 to 100 years, what</p><p>is the increase in the design range?) Data suggests that the exponent on n for</p><p>mR should be 0.72, which is presumably an influence of correlation.†</p><p>2.59. A proposed set of dam-operating policies ‡ can be characterized by</p><p>this (continuity) equation relating the reservoir storage at the end of the (i +</p><p>l)th time interval Si+1 to the storage at the end of the previous interval Si:</p><p>images</p><p>in which Xi+1 is the inflow during the (i + l)th interval and d is the “target”</p><p>draft (the “desired” amount of release) in any interval. The policy parameter</p><p>a (0 ≤ a ≤ 1) reflects the importance of carrying over water from one interval</p><p>to the next. If a = 0, Si+1 = 0 and there is no carryover; if a = 1, the policy is</p><p>the “normal release policy” (Prob. 2.32) (if, as we shall do, one neglects the</p><p>possibility of the available water in any year being less than d or more than d</p><p>+ c, where c is the dam’s capacity).</p><p>Assume that the inflows are uncorrelated with common mean mX and</p><p>variance images Our concern is to calculate the corresponding moments of</p><p>the storage S and the release or draft Y.</p><p>(a) Show that the mean and variance of the reservoir storage in any year</p><p>(along time after the initial year) are</p><p>images</p><p>Hint: the Si’s are not independent. Write an expression for Si in terms of the</p><p>independent Xi’s and take the limit as i becomes large. Check that ms and</p><p>images (valid presumably for both Si and Si+1 since the moments prove to</p><p>be independent of i) satisfy the equations found by taking the expectation</p><p>and variance, respectively, of both sides of the continuity equation. Discuss</p><p>the design implication of the variance for the “normal release policy,” a = 1.</p><p>(b) Show that the mean and variance of Y, the release in any year</p><p>(substantially after the first year), are</p><p>images</p><p>Hint: Show first that Yi+1 = (1 - a)(Xi+1 + Si) + ad.</p><p>Discuss the implications. If my were any other value, what would the</p><p>implications be? How do you explain the variance under the limiting cases,</p><p>a = 0 and a = 1?</p><p>(c) Show that the coefficient of correlation between Si and Si+1 is a. Discuss</p><p>this result for the limiting cases, in terms of one’s ability to predict Si+1</p><p>given Si.</p><p>(d) In the light of our assumption that we neglect the possibility of the</p><p>available water being very small or very large, discuss the conditions of</p><p>target draft and dam capacity for which the results above are valid.</p><p>Qualitatively, for a = 1, how would you expect these results to change if</p><p>either or both restrictions were dropped? Hint: See the sketch in Prob. 2.32.</p><p>You can easily prove to yourself that the continuity equation we have</p><p>adopted implies (for a = 1) that our operating policy is y = d for all z, which</p><p>may not be feasible.</p><p>2.60. The relationship below is often used in water-resources projects to</p><p>relate short-term benefits U to the amount of water released in any year F.†</p><p>(d is the announced or “target” release; see Prob. 2.32.)</p><p>(a) Find the annual expected benefits if the annual release Y has a uniform</p><p>distribution on the interval d/2 to 2d.</p><p>(b) Find the annual expected benefits when the annual release Y is related to</p><p>the available water Z by the “normal operating policy,” and when Z has the</p><p>distribution given in Prob. 2.32. Hint: If you do not have available the</p><p>distribution of Y (asked for in part (b) of Prob. 2.32) instead construct first a</p><p>functional relationship between U and Z.</p><p>images</p><p>Fig. P2.60</p><p>2.61. For a given target release d, the short-term benefits U of a dam release</p><p>Y are shown in the sketch with Prob. 2.60. If the planned or target release</p><p>were increased, however, long-term benefits associated with increased</p><p>downstream development (more irrigation, etc.) could accrue. In other</p><p>words, the value v associated with a release equal to the target value (see</p><p>sketch in Prob. 2.60) is itself an increasing function of d, the target release.</p><p>The combined result can be shown as indicated for two particular target</p><p>values d1 and d2.</p><p>images</p><p>Fig. P2.61</p><p>For a given capacity dam, assume that the distribution of F, the annual</p><p>release, is independent of d, the target release. The designer’s problem is to</p><p>choose that target release images. Show that the value of d that maximizes</p><p>the expected annual benefits is that value images such that†</p><p>images</p><p>In short, the optimal target value images is the value that divides the</p><p>density function of Y into two areas (f – h)/(g – h) to the left and (g – f)/(g –</p><p>h) to the right.</p><p>2.62. Moment generating and characteristic functions. The expectations</p><p>of two particular functions g(X) of random variables are of great value. They</p><p>are</p><p>images</p><p>and</p><p>images</p><p>in which i is the imaginary number images. The expectations of these</p><p>functions have special names and properties. The moment generating</p><p>junction (mgf) is defined, here for continuous variables, as</p><p>images</p><p>and the characteristic function (cf) as</p><p>images</p><p>(a) Show that these functions (integral transforms) can be used as “moment</p><p>generators.” In particular, show that</p><p>images</p><p>(b) Show that</p><p>images</p><p>(c) Show that for independent random variables X1 and X2 the moment</p><p>generating function and characteristic function of Y = X1 + X2 are</p><p>images</p><p>These simple relationships among mgf and cf of independent random</p><p>variables and their sums explains these functions’ great utility in modern</p><p>probability theory and mathematical statistics.</p><p>Although often difficult in practice, it is theoretically possible to transform</p><p>images back into fY(y). At a minimum the moments of Y are easily made</p><p>available through images</p><p>Hint: Because of the assumptions on convergence, the operations of</p><p>differentiation and expectation (integration) can be interchanged quite freely.</p><p>The student familiar with transform techniques in applied mathematics will</p><p>recognize these transforms and their relationships when he recalls Eq.</p><p>(2.3.43). We will not explore these transforms as they might be in the</p><p>remainder of the text, since most students are not familiar with their use.</p><p>(d) Find the moment generating function and characteristic function of the</p><p>random variable R in the illustration in Sec. 2.3.3:</p><p>images</p><p>Use these functions to find four moments of the variable. Hint: In</p><p>performing the integrations, one can assume u</p><p>the cost estimate of any bridge. But I am much more confident of our</p><p>estimate for the total cost of all four bridges because of the likelihood that a</p><p>high estimate on one will be balanced by a low estimate on another”</p><p>Discuss this statement. Use your knowledge, simply, of the means and</p><p>variances of sums of random variables to support your comments. If your</p><p>initial intuition lies with that of the spokesman, be sure you resolve in your</p><p>own mind why it is inconsistent. Compare both the variance and the</p><p>coefficients of variation of the total cost versus those of an individual cost.</p><p>If the four bridge sites are relatively close to one another, soil conditions,</p><p>although unknown, are probably similar. What implications does this</p><p>observation have for your analysis?</p><p>2.65 A total cost of earthwork on a road construction project will be the total</p><p>number of cubic yards excavated Y times the contractor’s unit bid price P. If</p><p>the mean and variance of the former are 100,000 yd3 and 10 X 106 (yd3)2,</p><p>and the mean and variance of P are 6 $/yd3 and 0.25 ($/yd3)2, respectively,</p><p>find the expected value and variance of the total cost, assuming that Y and P</p><p>are independent. Compare with the approximate values in Sec. 2.4.4. See</p><p>Prob. 2.52.</p><p>2.66. System reliability. A major application of probability has been in the</p><p>determination of the reliability of systems made up of components whose</p><p>reliabilities are known. (This reliability of a component is the probability</p><p>that it will function properly throughout the period of interest.) Simple block</p><p>diagrams are helpful in demonstrating how system performance depends on</p><p>component performance.</p><p>(a) Series system. For example, if a system will perform only if each and</p><p>every component proves reliable, then the block diagram will be chainlike:</p><p>images</p><p>Fig. P2.66a</p><p>If the events Ci = [component i performs satisfactorily] are independent,</p><p>show in terms of relationships among the events Ci that</p><p>images</p><p>if pi is the probability of images and (1 – ps) is the reliability of the total</p><p>system.</p><p>(b) Parallel redundant system. If the system will perform satisfactorily if</p><p>any one of the components “survives,” the block diagram is</p><p>images</p><p>Fig. P2.66b</p><p>For independent events Ci, show that the system reliability is</p><p>images</p><p>(c) Mixed systems. For a system indicated thus:</p><p>images</p><p>Fig. P2.66c</p><p>show that the system reliability, 1 – ps, is</p><p>images</p><p>for independent events Ci. Note that at the cost of an additional redundant</p><p>component, number 3, this system is more reliable than this simpler one.</p><p>images</p><p>Fig. P2.66d</p><p>(d) Examples. Nuclear power plants, which depend on the functioning of</p><p>many components, are designed with numerous redundancies. Assume, for</p><p>simplicity, that during a particular, major, design level of earthquake</p><p>intensity, the controlled shutdown of the reactor depends on the proper</p><p>functioning of the control system, the cooling system, and the primary</p><p>containment vessel. Assume that there are three redundant control systems,</p><p>two redundant cooling systems, and a single, steel primary containment with</p><p>two critical necessary components A and B. Block model the system and</p><p>calculate the system reliability with respect to shutdown, in terms of</p><p>component reliabilities. Assume independence of component performances</p><p>(at the given earthquake level).</p><p>There will be no major accident if either the shutdown is controlled or the</p><p>reinforced-concrete secondary containment vessel performs properly. Model</p><p>the total system with respect to major accident reliability.</p><p>(e) Component dependence. Recalculate the system reliability in part (c) if</p><p>the conditional probability of satisfactory performance of component 3 given</p><p>failure of component 2 is only images.</p><p>Dependence of component performance events is often introduced by the</p><p>systems environment or demand. If a large, random demand on the system is</p><p>the cause of the failure of component 2, it is likely to cause failure of</p><p>component 3 also.</p><p>Reconsider, for example, part (d), assuming now that the level of the</p><p>earthquake intensity is uncertain [not a given level as it was in (d)]. Then,</p><p>given an earthquake occurrence of uncertain level, the reliability of, say, the</p><p>secondary containment given failure of the primary vessel may be</p><p>substantially smaller than the (marginal) reliability of the secondary</p><p>containment. In the marginal analysis a whole spectrum of possible</p><p>earthquakes intensities had to be incorporated in the analysis. Failure of the</p><p>primary vessel suggests that a large intensity has probably occurred.</p><p>Conditional on this information, the reliability of the secondary containment</p><p>must be smaller.</p><p>2.67. In evaluating the consolidation settlement of the foundations of new</p><p>buildings, it is necessary to predict the sustained column loads that are</p><p>transmitted to the footing. We consider here only the sustained live loads.</p><p>Let the load on a particular column due to floor i be:</p><p>images</p><p>and that due to floor j be :</p><p>images</p><p>in which a is the tributary area. B is a random variable with a mean mB that</p><p>is equal to the average unit load over all buildings of this prescribed type of</p><p>use (e.g., offices), and a variance images, that is equal to the variance of</p><p>mean (over the building) building loads from building to building. Si and Sj</p><p>are random variables with zero mean representing the spatial variation of</p><p>load within a given building. They both have variance images. B and the</p><p>S’s are uncorrelated.</p><p>(a) What are the mean and variance of the total (sustained live) load</p><p>transmitted to a footing by a column supporting n such floors?</p><p>(b) For σB = 2σS and for σB = ½σs, sketch a plot of the coefficient of</p><p>variation of this total load versus n.</p><p>(c) What is the correlation coefficient between Lt- and Lj?</p><p>(d) What is the “partial correlation coefficient between Li and Lj, given that</p><p>B = b0? This coefficient is defined in the usual way, except that the</p><p>expectations are conditional on B = bQ. This coefficient appears again in</p><p>Sec. 4.3.1.</p><p>2.68. A storage reservoir is supplied with water at a constant rate k for a</p><p>period of time Y. Then water is drawn from it at the same rate for a period of</p><p>time X. X and Y are independent, with distributions</p><p>images</p><p>(Assume that the reservoir is infinitely large and contains an infinite amount</p><p>of water so that it cannot run dry or overflow.)</p><p>What are the probability density functions of Z = Y – X and of W = k(Y – X),</p><p>the change of the amount of water in the reservoir after one such cycle of</p><p>inflow and outflow?</p><p>2.69. Hazard functions. If an engineering system is subjected to a random</p><p>environment, its reliability can be defined in terms of the random variable T,</p><p>the time to failure, since the reliability of the system during a planned</p><p>lifetime of t0 is simply</p><p>Reliability (t0) = P[no failure before t0] = P[T > t0] = 1 – F(t0)</p><p>The hazard function h(t) is defined such that h(t) dt is the probability that the</p><p>failure will occur in the time interval t to t + dt given that no failure occurred</p><p>prior to time t.</p><p>(a) Show that</p><p>images</p><p>Its shape determines, for example, whether a system deteriorates with age or</p><p>wear (i.e., if h(t) grows with time).</p><p>(b) Show that the probability distribution of T, given a (“well-behaved,”</p><p>continuous) hazard function h(t), is</p><p>images</p><p>in which images</p><p>(c) Show that if the time to failure has an exponential distribution,</p><p>images</p><p>then the hazard function is a constant. In Sec. 3.2 we will call such</p><p>conditions “random” or Poisson failure events, and ν is their average rate of</p><p>arrival. It is a commonly adopted assumption in reliability analysis.</p><p>(d) Find the distribution of the time to failure T of a system which is</p><p>exposed to two independent kinds of hazard, one due to random occurrences</p><p>of “rare events” (e.g., earthquakes) and one due to wearout or deterioration</p><p>(e.g., fatigue). The rare events occur with average annual rate v. The hazard</p><p>due to wearout is negligible at time zero, but it grows linearly with time. At</p><p>time t = 10 years it is equal to that due to rare events. Justify why the</p><p>total</p><p>hazard function is just the sum of the two hazard functions. What is the</p><p>reliability of the system if it is desired that it operate for 20 years?</p><p>2.70. In simple frame structures (with rigid floors) such as the one shown in</p><p>the diagram, the total deformation of the top story Y is simply the sum of the</p><p>deformations of the individual stories X1 and X2, acting independently.</p><p>These variables are uncorrelated and have mean and variance m1 and</p><p>images, m2 and images, respectively.</p><p>(a). Find the mean and variance of Y.</p><p>(b). Find the correlation coefficient between Y and X2. Discuss the results in</p><p>terms of very large relative values of the moments (m1, m2, σ1, and σ2), for</p><p>example, σ2 much larger than σ1, and vice versa.</p><p>images</p><p>Fig. P2.70</p><p>(c) Find the correlation coefficient between Y and X2 if X1 and X2 are not</p><p>uncorrelated, but have a positive correlation coefficient p, owing, say, to the</p><p>common source of material and common constructor.</p><p>2.71. Tests on full-scale reinforced-concrete-bearing walls indicate that the</p><p>deflection of such a wall under a given horizontal load is a random variable.</p><p>The form of the distribution of the variable is</p><p>images</p><p>in which c = λk(k – 1)! for k integer. It has mean k/λ and variance k/λ2.</p><p>Different wall dimensions and different concrete properties will change the</p><p>values of the parameters k and λ.</p><p>In a small two-story building the deflection Y of the roof will be the sum of</p><p>the deflection of the first-story wall X and the deflection of the second-story</p><p>wall X2. X1 and X2 are assumed to be independent. Find the probability</p><p>density function of Y if</p><p>images</p><p>2.72. A harbor breakwater is made of massive tanks which are floated into</p><p>place over a shallow trench scraped out of the harbor floor and then filled</p><p>with sand. There is concern over the possibility of breakwater sliding under</p><p>the lateral pressure of a large wave in a major storm. It is difficult to predict</p><p>the lateral sliding capacity of such a system. What is the reliability (the</p><p>probability of satisfactory performance) of this system with respect to</p><p>sliding if the engineer judges the following?</p><p>(a) That the sliding resistance has a value 100, 120, or 140 units, with the</p><p>middle value twice as likely as the low value and twice as likely as the high</p><p>value.</p><p>(b) That the lateral force under the largest wave in the economic lifetime of</p><p>the breakwater X, has an exponential distribution with parameter λ = 0.02;</p><p>that is,</p><p>images</p><p>The units of sliding resistance and lateral force are the same. Resistance and</p><p>force are independent.</p><p>2.73. In planning a building, the number of elevators is chosen on a basis of</p><p>balancing initial costs versus the expected delay times of the users. These</p><p>delays are closely related to the number of stops the elevator makes on a</p><p>trip. If an elevator runs full (n people) and there are k floors, we want to find</p><p>the expected number of stops R the elevator makes on any trip. Assuming</p><p>that the passengers act independently and that any passenger chooses a floor</p><p>with equal probability 1/k, show that</p><p>images</p><p>Hint: It is often useful to define “indicator random variables” as follows. Let</p><p>Xi = 1 if the elevator stops at floor i, and 0 if it does not. Then observe that k</p><p>imagesFind the expected value of Xi after finding first the probability that</p><p>2.74. The peak annual wind velocity Xi in any year i at a certain site is often</p><p>assumed to have a distribution of the form</p><p>images</p><p>Peak annual wind velocities in different years are independent.</p><p>The pseudostatic force on an object subjected to the wind is proportional to</p><p>the square of the wind velocity: force = c(velocity)2. Find the probability</p><p>density function of Y, the maximum force on an object over a period of n</p><p>years.</p><p>2.75. The flooding (peak-flow) potential of a rain storm (defined here to be a</p><p>½-day period with more than 2 hr of rainfall) depends on both the total</p><p>rainfall Y and the duration X. Available data in a particular region suggests</p><p>that the PDF of X is approximately:</p><p>images</p><p>(a) Find k and sketch this PDF.</p><p>(b) The conditional distribution of the total rainfall F, given that the duration</p><p>X equals x hr, is uniformly distributed on the interval ½x – 1 to ½x + 2 in.:</p><p>images</p><p>What is the joint distribution of X and Y? Sketch it.</p><p>(c) “High” flow rates will occur if the rate of rainfall during the total storm</p><p>exceeds ⅔ in./hr. What fraction of storms cause “high” flow rates?</p><p>† If A is the certain event, i.e., if A is the collection of all sample points in</p><p>the sample space S, the complement Ac of event A will be the null event; i.e.,</p><p>it will contain no sample points.</p><p>‡ The square brackets [ ] should be read “the event that.”</p><p>† P[A] is read “the probability of the event A.”</p><p>‡ Upon returning in Sec. 2.2 to continuous sample spaces, where the number</p><p>of sample points is infinite, we shall not assign probabilities to specific</p><p>points but to small lengths or areas. Then the integral (sum) over all these</p><p>regions in the sample space must be unity.</p><p>† Such figures, which exploit the analogy between the algebra of events and</p><p>the areas in a plane, are called Venn diagrams. They can be most helpful in</p><p>visualizing event relationships, but it is important to realize that these</p><p>diagrams cannot illustrate relationships among numbers, that is, the</p><p>probabilities of events.</p><p>† Notice that the calculation of P[W2 | E10] on page 48, which is based on</p><p>the client’s four probability estimates, contradicts this assumption, for P[W2</p><p>| E10] = 0.86 does not equal P[W2] = 0.8. Hence, using the client’s</p><p>probabilities, the events are not independent. The engineer, on the other</p><p>hand, has assumed independence, and has estimated only P[E5] and P[W1],</p><p>and calculates the remaining probabilities of interest. The assumption of</p><p>independence reduces the number of numerical estimates necessary to</p><p>describe the phenomenon completely.</p><p>† In that chapter we shall find that, in general, decisions should be based on</p><p>utilities, which may or may not coincide with dollars.</p><p>† This and subsequent sections and illustrations marked with a dagger may</p><p>be omitted at first reading since they are of a more advanced nature.</p><p>‡ This example is based on the treatment developed in M. A. Benson and N.</p><p>C. Matalas [1965], The Effects of Floods and Obsolescence on Bridge Life,</p><p>Proc. ASCE, J. Highway Div., HW1, vol. 91, January.</p><p>† The engineering reader should take special care to read these equations in</p><p>terms of their event definitions. The following equation, for example, states</p><p>that the probability of neither a critical flood nor obsolescence in the first</p><p>year is the probability that no flood occurs multiplied by the probability that</p><p>no obsolescence occurs. In the more involved equations to follow,</p><p>understanding will usually come more quickly in this way than through</p><p>abstract symbols.</p><p>† This concept is discussed in detail in Sec. 3.1.</p><p>† Note that P[Ai | B400] is undefined, since P[B400] = 0 [see Eq (2.1.4a)].</p><p>† Since concrete strength increases with time, a strength of, say, 3500 psi</p><p>now suggests that the 28-day strength (which is used as a design standard)</p><p>was probably about 3000 psi.</p><p>† Unless, of course, the engineer excluded their possibility at the outset by</p><p>setting their prior probabilities to zero.</p><p>† More precisely, a random variable is a function denned on the sample</p><p>space of the experiment. It assigns a numerical value to every possible</p><p>outcome. (See, for example, Parzen [I960].)</p><p>‡ In general a capital letter will be used for a random variable, and the same</p><p>letter, in lowercase, will represent the values which it may take on.</p><p>† Two alternate ways of specifying a probability law—moment-generating</p><p>and characteristic functions—are discussed in Prob. 2.62, but will not be</p><p>used in this text.</p><p>† As will be seen, this does not imply that the random variable must take on</p><p>values over the entire axis. Intervals, such as the negative range, for</p><p>example, can be excluded (i.e., assigned zero probability).</p><p>† This line will usually be omitted, the PDF (or PMF or CDF) being tacitly</p><p>defined as equal to zero in regions where it is not specifically defined</p><p>(except that the CDF is 1 for values of the argument larger than the</p><p>indicated range).</p><p>† These functions are zero everywhere except at a single point where they</p><p>are infinite, although the integral under the “curve” is finite. They are</p><p>analogous to the concentrated load, pure impulse, or infinite source</p><p>employed elsewhere in civil engineering.</p><p>‡ Notice that in general some care must be taken with the inequalities. If a</p><p>jump does not occur in FX(x) at x, then the inequalities (≤ and ≥) and strict</p><p>inequalities () may be interchanged freely without altering the value</p><p>of the right-hand side of the equation. This is always the case if X is a</p><p>continuous random variable.</p><p>† Additional comment is made in Secs. 2.3.3, 3.3.1, and 3.5.3 on this</p><p>particular runoff problem.</p><p>† A more commonly adopted distribution for maximum wind velocities will</p><p>be discussed in Sec. 3.3.3.</p><p>† Note that for continuous random variables, GX(x) = P[X ≥ x] and for</p><p>discrete random variables, GX(x) = P[X > x].</p><p>† As mentioned in Sec. 2.2.1, the reader should remember that the</p><p>engineer’s mathematical model is conceptually different in kind from</p><p>observed data, histograms, scattergrams, observed relative frequencies, and</p><p>the like. In situations such as this, however, it may be quite reasonable</p><p>simply to adopt a model with probability assignments equal in numerical</p><p>value to previously observed relative frequencies.</p><p>† The theoretical difficulties here revolve around the fact that the probability</p><p>that Y takes on any specific value is zero. See, for example, Parzen [I960].</p><p>† Only continuous random variables are illustrated here, but in fact</p><p>continuous, discrete, and mixed random variables might well appear</p><p>simultaneously as jointly distributed random variables.</p><p>‡ Although rare in practice, the reader should be aware that pairwise</p><p>independence of random variables (for example, X and Y, X and Z, and Y and</p><p>Z) does not necessarily imply mutual independence (X, Y, and Z). Problem</p><p>2.18 demonstrates the parallel fact for events.</p><p>† It is important not to confuse the notions of functional dependence (the</p><p>more familiar) and stochastic or probabilistic dependence (Secs. 2.1.3 and</p><p>2.2.2). The first implies the second, but the converse is not true. In fact,</p><p>functional dependence can be thought of as “perfect” stochastic dependence.</p><p>If Y is functionally related to X by the equation Y = g(X), the conditional</p><p>distribution of Y given that X = x is a unit mass at y = g(x) and zero</p><p>elsewhere (assuming g(x) is single valued). That is, if there is functional</p><p>dependence, the joint probability density contours (Fig. 2.2.13) “squeeze”</p><p>together and merge into the line y= g(x).</p><p>† Such a function is said to be a “monotonically increasing” function.</p><p>‡ Such a function is said to express a “one-to-one” transformation between</p><p>the variables x and y. Note, for example, that the relationship between X and</p><p>Y in the preceding dispatching example is not one-to-one. Y = 2 could result</p><p>from X equal to either 4 or 5.</p><p>† This is a confusing, but standard, notation consistent with sin–1 (u) for the</p><p>arcsine or inverse sine function. In general, g–1(u) does not equal 1/g(u).</p><p>‡ An additional restriction, continuity of y = g(x), is also necessary for this</p><p>next step.</p><p>† This result is true on the average. In fact, for any given time T = t, the</p><p>population Q may better be considered a random variable. Such a model is</p><p>an example of a random function of t or a “stochastic process.’’ Note that</p><p>the population size, actually an integer number, is being treated for</p><p>convenience as a continuous variable, since it is a large number.</p><p>† For example, this is a common engineering assumption in the case where</p><p>X and Y represent the largest river flows in each of 2 successive years.</p><p>† For a discussion of this question see, for example, Rosenblueth [1964].</p><p>‡ This problem is considered in Esteva and Rosenblueth [1964]. For</p><p>Southern California it has been found empirically that c1 = ln 280/ln 2, c2 =</p><p>1/ln 2, and c3 = 1.7/ln 2 with R in kilometers. This empirical relationship is</p><p>not valid for small values of r (less than about 40 km), but it will be retained</p><p>here for simplicity.</p><p>† The reader is referred, however, to any of several texts (e.g., Freeman</p><p>[1963], page 82) for a particularly efficient technique for obtaining the joint</p><p>PDF of n random variables which are (simultaneous) functions of n other</p><p>jointly distributed random variables whose PDF is known; the method</p><p>applies to the case where the functional relationship is one-to-one and</p><p>continuous. It is sometimes called the method of Jacobeans. Equation</p><p>(2.3.12) is a special case when n = 1.</p><p>‡ This section can be omitted at first reading.</p><p>§ It should be noted that probabilistic Monte Carlo methods are also used to</p><p>evaluate multidimensional integrals which arise from problems which are</p><p>not probabilistic at all. (See Hammersly and Hanscomb [I960].)</p><p>¶ There is a negligible error here; as described, pN(n) does not equal 1 unless</p><p>n = 1, 2, . . ., ∞. See Sec. 3.2, where the distribution will be introduced as</p><p>the “Poisson distribution.” The total probability mass of the neglected terms</p><p>is small in this case.</p><p>† One of many schemes is described in Prob. 2.53.</p><p>† Or better, some kind of average value, say the centroid of the area under</p><p>the probability density curve.</p><p>† There can be as many intervals as there are members in the set of random</p><p>numbers from which the generator is selecting, and this number is</p><p>indefinitely large, since successive one-digit numbers can always be strung</p><p>together to make higher-order numbers. If decimal fractions between 0 and 1</p><p>(obtained by dividing the random number by the total number members in</p><p>the set) are employed, in the limit one obtains a continuous, uniform</p><p>distribution of the random numbers on the interval 0 to 1.</p><p>† This method implicitly makes use of the so-called “probability integral</p><p>transform,” Prob. 2.34.</p><p>† As we shall see in Chap. 4, the sample mean and sample variance (Sec.</p><p>1.2) also have significance with respect to the random variable T. They are</p><p>estimates of the first two moments (Sec. 2.4.1) of T.</p><p>† For a more formal statement of this, the law of large numbers, see Prob.</p><p>2.43.</p><p>† These two descriptors of a random variable, namely, the “most probable”</p><p>value, that at the peak of its PDF, and the “midpoint,” the value of x at which</p><p>the CDF equals ½, are known, respectively, as the mode and the median.</p><p>They are only seldom used (see Sec. 3.3.2 for one use) and will not be</p><p>considered further here. Clearly the mean, mode, and median coincide if a</p><p>distribution has a single peak and is symmetrical. Note, however, that</p><p>neither mode nor median may have a unique value for some distributions.</p><p>‡ This notion can be made more explicit through the law of large numbers,</p><p>Prob. 2.43.</p><p>† In the analysis and development of models, on the contrary, the variance is</p><p>a more fundamental notion and also is easier to work with, as will be seen in</p><p>subsequent sections.</p><p>† Since negative values are not meaningful in this problem, the lower two-</p><p>sigma bound on this variable is given as 0 rather than –233.</p><p>‡ The proof of this inequality is elementary and is outlined in a hint to Prob.</p><p>2.42.</p><p>† If only these first two conditions are assumed, for example, h2 can be</p><p>replaced by 2.25h2 in Eq. (2.4.11) (Freeman [1963]).</p><p>† No proof will be attempted here. It is hoped that the student can mentally</p><p>generalize upon the discrete example which is to follow well enough to</p><p>convince himself of the plausibility of the result.</p><p>‡ The expectation is said to “exist” if and only if the integral in Eq. (2.4.14)</p><p>is absolutely convergent, that is, if images</p><p>§ In this section, subsequent definitions will be stated only in terms of</p><p>continuous random variables. The extension to the discrete case will be</p><p>obvious.</p><p>† (γ2 – 3) is called the coefficient of excess. The value of 3 is chosen only</p><p>because it is the value of the kurtosis coefficient of a particular, commonly</p><p>used distribution (see Sec. 3.3.1).</p><p>† This is true unless one employs Dirac delta functions, as suggested in Sec.</p><p>2.2.1.</p><p>† Despite the superficial appearance of linear and quadratic relationships,</p><p>g1(X) and g2(y), and the implication that the mean and variance of X would</p><p>suffice to determine E[C], it should be noted that in fact these functions are</p><p>not linear or quadratic owing to their varying definitions over various ranges</p><p>(see Fig. 2.4.6b).</p><p>† If X and Y have the same dimensions, the dimensions of σX,Y are the same</p><p>as those of images; nonetheless convention dictates the notation σX,Y</p><p>rather than images</p><p>† One side of this contention is easily verified by assuming that Y = a + bX</p><p>and showing that ρ = 1 for b > 0 and – 1 for b</p><p>will appear in this chapter, and the</p><p>technique of comparison will be more fully discussed in Sec. 4.5. †</p><p>Such an empirical path is often the only one available to the engineer, but it</p><p>should be remembered that in these cases the subsequent conclusions, being</p><p>based on the adopted distribution, are usually dependent to a greater or</p><p>lesser degree upon its properties. In order to make the conclusions as</p><p>accurate and as useful as possible, and in order to justify, when necessary,</p><p>extrapolation beyond available data, it is preferable that the choice of</p><p>distribution be based whenever possible on an understanding of how the</p><p>physical situation might have given rise to the distribution. For this reason</p><p>this chapter is written to give the reader not simply a catalog of common</p><p>distributions but rather an understanding of at least one mechanism by which</p><p>each distribution might arise. To the engineer, the existence of such</p><p>mechanisms, which may describe a physical situation of interest to him, is</p><p>fundamentally a far more important reason for gaining familiarity with a</p><p>particular common distribution than, say, the fact that it is a good</p><p>mathematical approximation of some other distribution, or that it is widely</p><p>tabulated.</p><p>From the pedagogical point of view this chapter has the additional purpose</p><p>of reinforcing all the basic theory presented in Chap. 2. After the statement</p><p>of the underlying mechanism, the derivation of the various distributions</p><p>involved is only an application of those basic principles. In particular,</p><p>derived distribution techniques will be applied frequently in this chapter.</p><p>3.1 MODELS FROM SIMPLE DISCRETE RANDOM</p><p>TRIALS</p><p>Perhaps the single most common basic situation is that where the outcomes</p><p>of experiments can be separated into two exclusive categories— for</p><p>example, satisfactory or not, high or low, too fast or not, within</p><p>specifications or not, etc. The following distributions arise from random</p><p>variables of interest in this situation.</p><p>3.1.1 The Single Trial: The Bernoulli Distribution</p><p>We are interested in the simplest kind of experiment, one where the</p><p>outcomes can be called either a “success” or a “failure”; that is, only two</p><p>(mutually exclusive, collectively exhaustive) events are possible outcomes.</p><p>Examples include testing a material specimen to see if it meets</p><p>specifications, observing a vehicle at an intersection to see if it makes a left</p><p>turn or not, or attempting to load a member to its predicted ultimate capacity</p><p>and recording its ability or lack of ability to carry the load.</p><p>Although at the moment it may hardly seem necessary, one can define a</p><p>random variable on the experiment so that the variable X takes on one of two</p><p>numbers—one associated with success, the other with failure. The value of</p><p>this approach will be demonstrated later. We define, then, the “Bernoulli”</p><p>random variable X and assign it values. For example,</p><p>images</p><p>The choice of values 1 and 0 is arbitrary but useful. Clearly, the PMF of X is</p><p>simply</p><p>images</p><p>where p is the probability of success. The random variable has mean</p><p>images</p><p>and variance</p><p>images</p><p>As mentioned, another choice of values other than 0,1 might have been</p><p>made, but this particular choice yields a random variable with an expectation</p><p>equal to the probability of success p.</p><p>Thus, if one has a large stock of items, say high-strength bolts, and he</p><p>proposes to select and inspect one to determine whether it meets</p><p>specifications, the mean value of the Bernoulli random variable in such an</p><p>experiment (with a “success” defined as finding an unsatisfactory bolt) is p,</p><p>the proportion of defective bolts, † This direct correspondence between mX</p><p>and p will prove useful when in Chap. 4 we discuss methods for estimating</p><p>the mean value of random variables. Notice that the variance of X is a</p><p>maximum when p is ½.</p><p>3.1.2 Repeated Trials: The Binomial Distribution</p><p>A sequence of simple Bernoulli experiments, when the outcomes of the</p><p>experiments are mutually independent and when the probability of success</p><p>remains unchanged, are called Bernoulli trials. One might be interested, for</p><p>example, in observing five successive cars, each of which turns left with</p><p>probability p; under the assumption that one driver’s decision to turn is not</p><p>affected by the actions of the other four drivers, the set of five cars’ turning</p><p>behaviors represents a sequence of Bernoulli trials. From a batch of concrete</p><p>one extracts three test cylinders, each with probability p of failing at a stress</p><p>below the specified strength; these represent Bernoulli trials if the</p><p>conditional probability that any one will fail remains unchanged given that</p><p>any of the others has failed. In each of a sequence of 30 years, the</p><p>occurrence or not of a flood greater than the capacity of a spillway</p><p>represents a sequence of 30 Bernoulli trials if maximum annual flood</p><p>magnitudes are independent and if the probability p of an occurrence in any</p><p>year remains unchanged throughout the 30 years, †</p><p>We shall be interested in determining the distributions of various random</p><p>variables related to the Bernoulli trials mechanism.</p><p>Distribution Let us determine the distribution of the total number of</p><p>successes in n Bernoulli trials, each with probability of success p. Call this</p><p>random number of successes Y. Consider first a simple case, n = 3. There</p><p>will be no successes (that is, Y = 0) in three trials only if all trials lead to</p><p>failures. This event has probability</p><p>images</p><p>Any of the following sequences of successes 1 and failures 0 will lead to a</p><p>total of one success in three trials:</p><p>images</p><p>Each sequence is an event with probability of occurrence p(l – p)2.</p><p>Therefore the event Y = 1 has probability 3p(1– p)2, since the sequences are</p><p>mutually exclusive events. Similarly, the mutually exclusive sequences</p><p>images</p><p>each occuring with probability p2(l – p), lead to Y = 2. Hence</p><p>images</p><p>Similarly, P[Y = 3] = p3, since only the sequence 1, 1, 1 corresponds to Y =</p><p>3. In concise notation,</p><p>images</p><p>where images indicates the binomial coefficient:</p><p>images</p><p>This coefficient is equal to the number of ways that exactly y successes can</p><p>be found in a sequence of three trials. (Recall that 0! = 1, by definition.)</p><p>If the probability that a contractor’s bid will be low is ½ on each of three</p><p>independent jobs, the probability distribution of Y, the number of successful</p><p>bids, is</p><p>images</p><p>Contrary to some popular beliefs, it is not certain that the contractor will get</p><p>a job; the likelihood is almost ⅓ that he will get no job at all!</p><p>In general, if there are n Bernoulli trials, the PMF of the total number of</p><p>successes Y is given by</p><p>images</p><p>images</p><p>It is clear that the parameter n must be integer and the parameter p</p><p>must be 0 ≤ p ≤ 1. The binomial coefficient</p><p>images</p><p>is the number of ways a sequence of n trials can contain exactly y successes</p><p>Parzen [I960].</p><p>A number of examples of this, the “binomial distribution,’ are plotted in Fig.</p><p>3.1.1. The shape depends radically on the values of the parameters n and p.</p><p>Notice the use of the notation B(n,p) to indicate a binomial distribution with</p><p>parameters n and p; for example, the random variable Y in the bidding</p><p>example above is B(3,⅓).</p><p>images</p><p>Fig. 3.1.1 Binomial distribution B(n,p).</p><p>Formally, the CDF of the binomial distribution is</p><p>images</p><p>Thus in the bidding example, the probability that the contractor will receive</p><p>two or fewer bids is</p><p>images</p><p>Or, more easily,</p><p>images</p><p>It is often the case that the probabilities of complementary events are more</p><p>easily computed than those of the events themselves. As another example,</p><p>the probability that the contractor gets at least one bid is</p><p>images</p><p>Moments The mean and variance of a binomial random variable Y are easily</p><p>determined, and the exercise permits us to demonstrate some of the</p><p>techniques of dealing with sums which many readers find unfamiliar. Thus,</p><p>images</p><p>Since the first term is zero, it can be dropped. The y cancels a term in y!:</p><p>images</p><p>Bringing np in front of the sum,</p><p>images</p><p>Letting u = y – 1,</p><p>images</p><p>Notice now that the sum is over</p><p>all the elements of a B(n – 1, p) mass</p><p>function and hence equals 1, yielding</p><p>images</p><p>This is an example of a most common method of approach in probability</p><p>problems, namely, manipulating integrals and sums into integrals and sums</p><p>over recognizable PDF or PMF’s, which are known to equal unity. In a</p><p>similar manner, we can find</p><p>images</p><p>or</p><p>images</p><p>Y as a sum Notice that the total number of successes in n trials Y, can be</p><p>interpreted as the sum</p><p>images</p><p>of n independent identically distributed Bernoulli random variables; thus Xi</p><p>= 1 if there is a success on the ith trial, and Xi = 0 if there is a failure. As</p><p>such, the results above could have been more easily obtained as</p><p>images</p><p>using Eqs. (3.1.2) and (3.1.3). The techniques of derived distributions (Sec.</p><p>2.3) could also have been used to find the distribution of Y from those of Xi</p><p>The interpretation of a binomial variable as the sum of Bernoulli variables</p><p>also explains why the sum of two binomial random variables, B(n1,p) and</p><p>B(n2,p), also has a binomial distribution, B(n1 + n2, p) as long as p remains</p><p>constant. This fact is easily verified using the techniques of Sec. 2.3.</p><p>The binomial distribution is tabulated, † but its evaluation, which is time-</p><p>consuming for large n, can frequently be carried out approximately using</p><p>more readily available tables of the Poisson distribution (Sec. 3.2.1) or the</p><p>normal distribution (Sec. 3.3.1); Prob. 3.18 explains these approximations.</p><p>The latter approximation is justified in Sec. 3.3.1 in the light of the binomial</p><p>random variable’s interpretation as the sum of n Bernoulli random variables.</p><p>3.1.3 Repeated Trials: The Geometric and Negative Binomial</p><p>Distributions</p><p>Rather than asking the question, “How many successes will occur in a fixed</p><p>number of repeated Bernoulli trials?” the engineer may alternatively be</p><p>interested in the question, “At what trial will the first success occur?” For</p><p>example, how many bolts will be tested before a defective one is</p><p>encountered if the proportion defective is p? When will the first critical</p><p>flood occur if the probability that it will occur in any year is p?</p><p>Geometric distribution Assuming independence of the trials and a constant</p><p>value of p, the distribution of N, the number of trials to the first success, is</p><p>easily found. The first success will occur on the nth trial if and only if (1) the</p><p>first n – 1 are failures, which occurs with probability (1 – p)n–1 and (2) the</p><p>nth trial is a success, which occurs with probability p, or</p><p>images</p><p>Notice that N may, conceptually at least, take on any value from one to</p><p>infinity. ‡</p><p>This distribution is called the geometric § distribution with parameter p.</p><p>Symbolically we say that N is G(p). A plot of a geometric distribution</p><p>appears in Fig. 3.1.2.</p><p>The cumulative distribution function of the geometric distribution is</p><p>images</p><p>images</p><p>Fig. 3.1.2 Geometric distribution G(p).</p><p>where use is made of the familiar formula for the sum of a geometric</p><p>progression. This result could have been obtained directly by observing that</p><p>the probability that N ≤ n is simply the probability that there is at least one</p><p>occurrence in n trials, or</p><p>images</p><p>The geometric distribution also follows from a totally different mechanism</p><p>(see Sec. 3.5.3).</p><p>Moments of the geometric The first moment of a geometric random</p><p>variable is found by substitution and a number of algebraic manipulations</p><p>similar to those used in the binomial case. The reader can show that</p><p>images</p><p>In words, the average number of trials to the first occurrence is the</p><p>reciprocal of the probability of occurrence on any one trial. For example, if</p><p>the proportion of defective bolts is 0.1, on the average, 10 bolts would be</p><p>tested before a defective would be found, assuming the conditions of</p><p>independent Bernoulli trials hold.</p><p>The variance of N can easily be shown to be</p><p>images</p><p>Negative binomial distribution We have just determined the answer to the</p><p>question, “On which trial will the first success occur?” Next consider the</p><p>more general question, “On which trial will the kth success occur?” We are</p><p>dealing now with a random variable, say Wk, which is the sum of random</p><p>variables N1N2, . . ., Nk, where Ni is the number of trials between the (i – l)th</p><p>and ith successes. Thus</p><p>images</p><p>Because of the assumed independence of the Bernoulli trials, it is clear that</p><p>the Ni: (i = 1,2, . . . k) are mutually independent random variables, each with</p><p>a common geometric distribution with parameter p. As a result of this</p><p>observation, we can obtain partial information about Wk, namely, its</p><p>moments, very easily. The mean and variance of Wk can be written down</p><p>immediately as</p><p>images</p><p>images</p><p>The distribution of W can be found using the methods discussed in Sec. 2.3.</p><p>For example, for k = 2,</p><p>images</p><p>Writing Eq. (2.3.43) for the discrete case, we have</p><p>images</p><p>which states that the probability that W2 = w is the probability that N1= n</p><p>and N2= w – n, summed over all values of n up to w. Notice that when n = w</p><p>in the sum, we get images, which is zero; hence</p><p>images</p><p>or, simply,</p><p>images</p><p>This result could have been arrived at directly by arguing that the second</p><p>success will be achieved at the wth trial only if there are w – 2 failures, a</p><p>success on the wth trial, and one success in any one of the w – 1 preceding</p><p>trials.</p><p>With this result for W2, one could find the distribution of W3, knowing W3 =</p><p>W2+ N3. In turn the distribution for any k could be arrived at. If this exercise</p><p>is carried out, a pattern emerges immediately, and one can conclude that</p><p>images</p><p>The result is reasonable if one argues that the kth success occurs on wth trial</p><p>only if there are exactly k – 1 successes in the preceding w – 1 trials and</p><p>there is a success on the wth trial. The probability of exactly k – 1 successes</p><p>in w – 1 trials we know from our study of binomial random variables to be</p><p>images</p><p>This distribution is known as the Pascal, or negative binomial, distribution</p><p>with parameters k and p, denoted here NB(k,p).† It has been widely used in</p><p>studies of accidents, ‡ cosmic rays, learning processes, and exposure to</p><p>diseases. It should not be surprising, owing to its possible interpretation as</p><p>the number of trials to the (k1 + k2)th event, that a random variable which is</p><p>the sum of a NB(k1,p) and a NB(k2,p) pair of (independent) random variables</p><p>is itself negatively binomially distributed, NB(k1 + k2, p).</p><p>Illustration: Turn lanes If a left-turn lane at a traffic light has a capacity of</p><p>three cars, what is the probability that the lane will not be sufficiently large</p><p>to hold all the drivers who want to turn left in a string of six cars delayed by</p><p>a red signal? In the mean, 30 percent of all cars want to turn left. The desired</p><p>probability is simply the probability that the number of trials to the fourth</p><p>success (left-turning car) is less than 6. Letting WA equal that number, W4</p><p>has a negative binomial distribution. Therefore,</p><p>images</p><p>Alternatively, one could have deduced this answer by asking for the</p><p>probability of four, five, or six successes in six Beroulli trials and used the</p><p>PMF of the appropriate binomial random variable.</p><p>A more realistic question might be, “What is the probability that the</p><p>left-turn lane capacity will be insufficient when the number of red-signal-</p><p>delayed cars is unspecified?” This number is itself a random variable, say Z,</p><p>with a probability mass function, pz(z), z = 0, 1, 2, .... Then the probability</p><p>of the event A, that the lane is inadequate, is found by considering all</p><p>possible values of Z:</p><p>images</p><p>or, since Fw4(z)= 0 for z</p><p>which the design can</p><p>withstand, e.g., the maximum flow possible through a spillway or the</p><p>maximum wind velocity which a structure can resist. The engineer then</p><p>seeks (from past data, say) an estimate of the probability p that in a given</p><p>time period, usually a year, this critical magnitude will be exceeded.</p><p>If, as is commonly assumed, the magnitude of the maximum annual</p><p>flow rates in a river or the maximum annual wind velocities are independent,</p><p>and if p remains constant from year to year, then the successive years</p><p>represent independent Bernoulli trials. A magnitude greater than the critical</p><p>value is denoted a success. With the knowledge of the distributions in this</p><p>section the engineer is now in a position to answer a number of critical</p><p>questions.</p><p>Let us assume, for example, that p = 0.02; that is, there is 1 chance</p><p>in 50 that a flood greater than the critical value will occur in any particular</p><p>year. What is the probability that at least one large flood will take place</p><p>during. the 30-year economic lifetime of a proposed flood-control system?</p><p>Let X equal the number of large floods in 30 years. Then X is B(30,0.02),</p><p>and</p><p>images</p><p>Using the familiar binomial theorem,</p><p>images</p><p>If this risk of at least one critical flood is considered too great</p><p>relative to the consequences, the engineer might increase the design capacity</p><p>such that the magnitude of the critical flood would only be exceeded with</p><p>probability 0.01 in any one year. Then X is B(30,0.01), and</p><p>images</p><p>The engineer seeks, of course, to weigh increased initial cost of the</p><p>system versus the decreased risk of incurring the damage associated with a</p><p>failure of the system to contain the largest flood.</p><p>The number of years N to the first occurrence of the critical flood is a</p><p>random variable with a geometric distribution, G(0.01), in the latter case.</p><p>The probability that it is greater than 10 years is</p><p>images</p><p>What is the probability that N > 30? Clearly it is just equal to the</p><p>probability that there are no floods in 30 years, i.e., that X = 0, where X is</p><p>B(30,0.01). Here, using a previous result,</p><p>images</p><p>Average Return Periods The expected value of N is simply</p><p>images</p><p>This is the average number of trials (years) to the first flood of</p><p>magnitude greater than the critical flood. In civil engineering this is referred</p><p>to as the average return period or simply the return period. The critical</p><p>magnitude is often referred to as the “mN-year flood,” here the “100-year</p><p>flood.” The term is somewhat unfortunate, since its use has led the layman</p><p>to conclude that there will be 100 years between such floods when in fact</p><p>the probability of such a flood in any year remains 0.01 independently of the</p><p>occurrence of such a flood in the previous or a recent year (at least</p><p>according to the engineer’s model).</p><p>The probability that there will be no floods greater than the m-year</p><p>flood in m years is, since X is then B[m,l/m],</p><p>images</p><p>where u = m(l/m) = 1; hence, for large m,</p><p>images</p><p>That is, the likelihood that one or more m-year events will occur in m</p><p>years is approximately 1 – e–1= 0.632. Thus a system “designed for the m-</p><p>year flood” will be inadequate with a probability of about 2/3 at least once</p><p>during a period of m years.</p><p>Cost Optimization What is the optimum design to minimize total expected</p><p>cost? Assume that the cost c associated with a system failure is independent</p><p>of the flood magnitude, and that, in the range of designs of interest, the cost</p><p>is a constant I plus an incremental cost whose logarithm is proportional to</p><p>the mean return period mN of the design demand, † For an economic design</p><p>life of 30 years,</p><p>images</p><p>Since the cost of failure is cX,</p><p>images</p><p>Assume that the system itself remains in use with an unaltered capacity</p><p>after any failure. Then, if X is the number of failures in 30 years, it is</p><p>distributed</p><p>B(30, 1/mN) (assuming no more than one failure per year). The mean of</p><p>X is 30/mN. The expected cost, given a value of mN, is</p><p>images</p><p>The design flood magnitude images, which minimizes this expected</p><p>total cost, is found by setting the derivative of total cost equal to zero:</p><p>images</p><p>This equation can be solved by trial and error for given values of b and</p><p>c. For example, for b = 0.1 year-1 and c = $200,000, images years. Thus</p><p>the designer should provide a system with a capacity sufficient to withstand</p><p>the demand which is exceeded with probability 1/90 = 0.011 in any year. In</p><p>Sec. 3.3, we will encounter distributions commonly used to describe annual</p><p>maximum floods from which such design magnitudes could be easily</p><p>obtained, once the design return period is fixed.</p><p>3.1.4 Summary</p><p>The basic model discussed here is that of Bernoulli trials. They are a</p><p>sequence of independent experiments, the outcome of any one of which can</p><p>be classified as either success or failure (or, numerically, 0 or 1). The</p><p>probability of success p remains constant from trial to trial.</p><p>If this model holds, then:</p><p>1. Y, the total number of successes in n trials, has a binomial distribution:</p><p>images</p><p>with mean np and variance np(1 – p).</p><p>2. N, the trial number at which the first success occurs, has a</p><p>geometricdistribution:</p><p>images</p><p>with mean 1/p and variance (1 – p)/p2. The former is called the mean</p><p>return period.</p><p>3. Wk, the trial number at which the kth success will occur, has a negative</p><p>binomial distribution:</p><p>images</p><p>with mean k/p and variance k (l – p)/p2.</p><p>3.2 MODELS FROM RANDOM OCCURRENCES</p><p>In many situations it is not possible to identify individual discrete trials at</p><p>which events (or successes) might occur, but it is known that the number of</p><p>such trials is many. The models in this section arise out of consideration of</p><p>situations where the number of possible trials is infinitely large. Examples</p><p>include situations where events can occur at any instant over an interval of</p><p>time or at any location along the length of a line or on the area of a surface.</p><p>3.2.1 Counting Events; The Poisson Distribution</p><p>A derivation of the distribution Suppose that a traffic engineer is</p><p>interested in the total number of vehicles which arrive at a specific location</p><p>in a fixed interval of time, t sec long. If he knows that the probability of a</p><p>vehicle occurring in any second is a small number p, (and if he assumes that</p><p>the probability of two or more vehicles in any one second is negligible), then</p><p>the total number of vehicles X in the n = t (assumed) independent trials is</p><p>binomial, B(n,p):</p><p>images</p><p>Consider what happens as the engineer takes smaller and smaller time</p><p>durations to represent the individual trials. The number of trials n increases</p><p>and the probability p of success on any one trial decreases, but the expected</p><p>number events in the total interval must remain constant at pn. Call this</p><p>constant ν and consider the PMF of X in the limit as the trial duration</p><p>shrinks to zero, such that</p><p>images</p><p>Substituting for p = ν/n in the PMF of X and rearranging,</p><p>images</p><p>The term in braces has x terms in the numerator and x terms in the</p><p>denominator. For large n each of these terms is very nearly equal to n; hence</p><p>in the limit, as n goes to infinity, the term in braces is simply nx/nx or 1. The</p><p>term (1 – ν/n)n is known to equal e–ν in the limit. Hence the PMF of X is</p><p>images</p><p>Moments This extremely useful distribution is known as the Poisson</p><p>distribution, denoted here P(ν). Notice it contains but a single parameter ν,</p><p>compared with the two required for the binomial distribution, B(n,p). Its</p><p>mean and variance are both equal to this parameter. Following the same</p><p>steps used to find the mean of the binomial distribution,</p><p>images</p><p>Letting y = x – 1,</p><p>images</p><p>since the sum is now simply the sum over a Poisson PMF. A similar</p><p>calculation shows that also</p><p>images</p><p>Plots of Poisson distributions are displayed in Fig. 3.2.1. † Notice the fading</p><p>of skew as ν increases.</p><p>Also, consideration of the derivation should make it clear that the sum</p><p>In this case, if the students reported their experimental results in the form of</p><p>only three numbers, images, s, and g1, it would already be sufficient to</p><p>gain appreciation for the shape of the histogram. The economy in the use of</p><p>such sample averages to transmit information about data becomes even</p><p>more obvious as the amount of data increases.</p><p>1.3 DATA OBSERVED IN PAIRS</p><p>If paired samples of two items of related interest, such as the first-crack</p><p>load and the failure load of a beam (Table 1.2.1), are available, it is often of</p><p>interest to investigate the correlation between them. A graphical picture is</p><p>available in the so-called scattergram, which is simply a plot of the</p><p>observed pairs of values. The scattergram of the reinforced-concrete–beam</p><p>data is shown in Fig. 1.3.1, where the xi are values of first-crack loads and</p><p>the yi are values of failure loads. There is no clearly defined functional</p><p>relationship between these observations, even though an engineer might</p><p>expect larger-than-average values of one load generally to pair with larger-</p><p>than-average values of the other, and similarly with low values.</p><p>A numerical summary of the tendency towards high-high, low-low pairings</p><p>is provided by the sample covariance sXY, defined by</p><p>images</p><p>Clearly, if larger (smaller) than average values of x are frequently paired</p><p>with larger (smaller) than average values of y, most of the terms will be</p><p>positive, while small-large pairings will tend to yield negative values of sX,</p><p>Y.</p><p>images</p><p>Fig. 1.3.1 Beam-data scattergram; plot shows lack of linear correlation.</p><p>It is common to normalize the sample covariance by the sample standard</p><p>deviations, denoted now with subscripts, sX and sY. The result is called the</p><p>sample correlation coefficient rX,Y:</p><p>images</p><p>It can be shown that rX,Y is limited in value to – 1 ≤ rX,Y ≤ 1 and that the</p><p>extreme values are obtained if and only if the points in the scattergram lie</p><p>on a perfectly straight line, that is, only if</p><p>images</p><p>the sign of rX,Y depending only on the sign of b. In this case the factors are</p><p>said to be perfectly correlated. For other than perfectly linear relationships</p><p>|rX,Y| is less than 1, the specific value rX,Y = 0 being said to indicate that the</p><p>x’s and y’s are uncorrelated. The x’s and y’s may, in fact, lie on a very well-</p><p>defined nonlinear curve and hence be closely, perhaps functionally, related</p><p>(for example, yi = bxi</p><p>2); in this case, the absolute value of the sample</p><p>correlation coefficient will be less than one. The coefficient is actually a</p><p>measure of only the linear correlation between the factors sampled.</p><p>For the beam data of Table 1.2.1, the sample covariance is while the sample</p><p>correlation coefficient is</p><p>images</p><p>images</p><p>Fig. 1.3.2 Scattergram of MLVS against MLSS.</p><p>images</p><p>The small value of this coefficient summarizes the qualitative conclusion</p><p>reached by observing the scattergram, that is, that the first-crack loads and</p><p>failure loads are not closely related. To the engineer who must judge the</p><p>ultimate strength of a (moment) cracked beam, the implications are that the</p><p>crack does not necessarily mean that (moment) failure is imminent, and</p><p>also that he cannot successfully use the first-crack load to help predict the</p><p>ultimate load better (Sec. 4.3).</p><p>Illustration: Large correlation As a second example consider a problem</p><p>in mixed-liquor analysis. Suspended solids (MLSS) can be readily</p><p>measured, but volatile solids (MLVS) prove more difficult. It would be</p><p>convenient if a measure of MLSS could be interpreted in terms of MLVS.</p><p>A scattergram between MLVS and MLSS is shown in Fig. 1.3.2. The</p><p>sample correlation coefficient for the data shown is rX,Y = 0.92. This</p><p>relatively large value of the correlation coefficient follows from the nearly</p><p>linear relationship between the two factors in the scattergram. It also</p><p>suggests that the value of the MLVS could be closely estimated given a</p><p>measurement of MLSS. Methods for carrying out and evaluating this</p><p>prediction will be investigated in Sec. 4.3.</p><p>Discussion In addition to providing efficient summaries of groups of</p><p>numbers, the data-reduction techniques discussed in this chapter play an</p><p>important role in the formulation and implementation of probability-based</p><p>engineering studies. The data in the illustrations and problems in this</p><p>chapter might be interpreted by the engineer as repeated observations of a</p><p>random or probabilistic mathematical model that he formulated to aid him</p><p>in describing and predicting some natural physical phenomenon of concern.</p><p>The concepts introduced in Chap. 2 are fundamental to the construction and</p><p>manipulation of these probabilistic models. In Chap. 4 we shall be</p><p>concerned with the use made of physical-world observed data, such as that</p><p>seen in this chapter, in fitting the mathematical models, and in verifying</p><p>their appropriateness.</p><p>The importance of the histogram in choosing and in verifying the</p><p>mathematical model will become evident in Chaps. 2 to 4. Parameters of</p><p>the mathematical models which are analogous to the sample mean, sample</p><p>variance, and sample correlation coefficient will be discussed in Sec. 2.4.</p><p>The relationship between sample averages and these model parameters will</p><p>be treated in detail in Chap. 4.</p><p>1.4 SUMMARY FOR CHAPTER 1</p><p>In Chap. 1 we introduced graphical and numerical ways to reduce sets of</p><p>observed data. The former include various forms of relative frequency and</p><p>cumulative frequency diagrams. The latter include measures of central</p><p>tendency, dispersion, skew, and correlation. The most commonly</p><p>encountered are the sample mean images, the sample standard deviation</p><p>s, the sample coefficient of variation υ, and the sample correlation</p><p>coefficient r.</p><p>REFERENCES</p><p>General</p><p>The student will find it desirable to refer to other texts to amplify this</p><p>chapter. In particular, Hald [1952] will prove most useful.</p><p>Bowker, A. H. and G. J. Lieberman [1959J: “Engineering Statistics,”</p><p>Prentice-Hall, Inc., Englewood Cliffs, N.J.</p><p>Bryant, E. C. [I960]: “Statistical Analysis,” McGraw-Hill Book Company,</p><p>New York.</p><p>Hahn, G. J. and S. S. Shapiro [1967]: “Statistical Models in Engineering,”</p><p>John Wiley & Sons, Inc., New York.</p><p>Hald, A. [1952]: “Statistical Theory with Engineering Applications,” John</p><p>Wiley & Sons, Inc., New York.</p><p>Neville, A. M. and J. B. Kennedy [1964]: “Basic Statistical Methods for</p><p>Engineers and Scientists,” International Textbook Company, Scranton, Pa.</p><p>Wine, R. L. [1964]: “Statistics for Scientists and Engineers,” Prentice-Hall,</p><p>Englewood Cliffs, N.J.</p><p>Specific text references</p><p>Sturges, H. A. [1926]: The Choice of a Class Interval, J. Am. Statist. Assoc,</p><p>vol. 21, pp. 65–66.</p><p>PROBLEMS</p><p>1.1. Ten timber beams were tested on a span of 4 ft by a single concentrated</p><p>load at midspan. The Douglas fir beams were 2 by 4 in. in nominal section</p><p>(actual section: 1.63 by 3.50 in.). The purpose of the study was to compare</p><p>ultimate load and load allowable by building code (394 lb), to compare</p><p>actual and calculated deflection (E = 1,600,000 psi), and to determine if a</p><p>relationship exists between rigidity and ultimate strength.</p><p>images</p><p>Compute the sample mean and variance for each set of data. Construct</p><p>histograms and relative frequency distributions for each set. Plot the scatter</p><p>diagram and compute the correlation coefficient. What are your</p><p>conclusions?</p><p>1.2. Ten bolted timber joints were tested using a ½-in. bolt placed in double</p><p>shear through three pieces of 2 by 4 in. (nominal size) Douglas fir. The</p><p>results of the tests were:</p><p>images</p><p>Calculate the sample standard deviations and coefficients of variation. Plot</p><p>a scatter diagram and compute the sample correlation coefficient between</p><p>load at yield of ⅛ in. and ultimate load. Might this simpler, nondestructive</p><p>test be considered as a prediction of ultimate joint capacity?</p><p>1.3. Jack and anchor forces were measured in a prestressed-concrete lift-</p><p>slab job for both straight and curved cables. The objective of the study was</p><p>to obtain estimates of the influence of friction and curvature. The data are</p><p>given in terms of the ratio of jack to</p><p>of</p><p>two Poisson random variables with parameters vx and v2 must again be a</p><p>Poisson random variable with parameters ν= ν1 + ν2- (How might this fact</p><p>be verified?) Distributions with the peculiar and valuable property that the</p><p>sum of independent random variables with the distribution has the same</p><p>distribution are said to be “regenerative.” The binomial and negative</p><p>binomial distributions are, recall, regenerative only on the condition that the</p><p>parameter p is the same for all the distributions.</p><p>Poisson process It is clear from the derivation that if a time interval of a</p><p>different duration, say 2t, is of interest, then the number of trials at any stage</p><p>in the limit would be twice as great and the parameter of the resulting</p><p>Poisson distribution would be 2ν. In such cases, the parameter of the Poisson</p><p>distribution can be written advantageously as λt rather than ν:</p><p>images</p><p>Fig. 3.2.1 Poisson distribution P(ν).</p><p>images</p><p>This form of the Poisson distribution is suggestive of its association with the</p><p>Poisson process. A stochastic process is a random function of time (usually).</p><p>In this case we are interested in a stochastic process X(t), whose value at any</p><p>time t is the (random) number of arrivals or incidents which have occurred</p><p>since time t = 0. Just as samples of a random variable X are numbers, x1, x2,</p><p>. . ., so observations of a random process X(t) are sample functions of time,</p><p>x1(t), x2(t), . . ., as shown in Fig. 3.2.2.</p><p>images</p><p>Fig. 3.2.2 Sample functions of a Poisson process X(t).</p><p>The samples shown there are of a Poisson process which is counting the</p><p>number of vehicle arrivals versus time. Other examples of stochastic</p><p>processes include wave forces versus time, total accumulated rainfall versus</p><p>time, and strength of soil versus depth. We will encounter other cases of</p><p>stochastic processes in Secs. 3.6 and 3.7 and in Chap. 6.</p><p>At any fixed value of the (time) parameter t, say t = t0, the value X(t0) of a</p><p>stochastic process is a simple random variable, with an appropriate</p><p>distribution images which has an appropriate mean images variance</p><p>images etc.† In general, this distribution, mean, and variance are functions</p><p>of time. In addition, the joint behavior of two (or more) values, say X(t0) and</p><p>X(t1), of a stochastic process is governed by a joint probability law.</p><p>Typically one might be interested in studying the conditional distribution of</p><p>a future value X(t1), given an observation at the present time X(t0), in order</p><p>to “predict” that future value (e.g., see Sec. 3.6.2).</p><p>An elementary result of the study of stochastic processes‡ is that the</p><p>distribution of the random variable X(t0) is the Poisson distribution with</p><p>parameter λt0 if the stochastic process X(t) is a Poisson process with</p><p>parameter λ. To be a Poisson process, the underlying physical mechanism</p><p>generating the arrivals or incidents must satisfy the following important</p><p>assumptions:</p><p>1. Stationarity. The probability of an incident in a short interval of time t to t</p><p>+ h is approximately λh, for any t.</p><p>2. Nonmultiplicity. The probability of two or more events in a short interval</p><p>of time is negligible compared to λh (i.e., it is of smaller order than λh).</p><p>3. Independence. The number of incidents in any interval of time is</p><p>independent of the number in any other (nonoverlapping) interval of time.</p><p>The close analogy to the assumptions underlying discrete Bernoulli trials</p><p>(Sec. 3.1) is evident. Therefore, the observed convergence of the binomial</p><p>distribution to the Poisson distribution is to be expected.</p><p>In short, if we are observing a Poisson process, the distribution of X(t) at any</p><p>t is the Poisson distribution [Eq. (3.2.4)], with mean mX(t)=λt [from Eq.</p><p>(3.2.2)]. Therefore, the parameter λ is usually referred to as the average rate</p><p>(of arrival) of the Poisson process.</p><p>The basic mechanism from which the Poisson process arises, namely,</p><p>independent incidents occurring along a continuous (time) axis with a</p><p>constant average rate of occurrence, suggests why it is often referred to as</p><p>the “model of random events” (or random arrivals). It has been successfully</p><p>employed to describe such diverse problems as the occurrences of storms</p><p>(Borgman [1963]), major floods (Shane and Lynn [1964]), overloads on</p><p>structures (Freudenthal, Garrelts, and Shinozuka [1966]), and, distributed in</p><p>space rather than time, flaws in materials and particles of aggregate in</p><p>surrounding matrices of material,† It is also widely employed in other fields</p><p>of engineering to describe the arrival of telephone calls at a central exchange</p><p>and the demands upon service facilities.</p><p>The Poisson process is often used by traffic engineers to model the flow of</p><p>vehicles past a point when traffic is freely flowing and not dense (e.g.,</p><p>Greenshields and Weida [1952], Haight [1963]). For this reason the Poisson</p><p>distribution describes well the number of vehicles which arrive at an</p><p>intersection during a given time interval, say a cycle of a traffic light. The</p><p>distribution logically might have been used in the illustrations dealing with</p><p>traffic lights in Secs. 2.3 and 3.1.</p><p>Illustration: Left-turn lane and the model of random selection If, in the</p><p>illustration in Sec. 3.1.2 involving left-turning cars at a traffic light, the</p><p>number Z of cars arriving during one cycle is Poisson-distributed with</p><p>parameter ν = 6, the</p><p>probability that less than four cars will arrive is</p><p>images</p><p>If this event occurs, there is no chance that the left-turn lane capacity</p><p>will be exceeded. The probability of the event A, that the lane will be</p><p>inadequate on any cycle (assuming there are no cars remaining from a</p><p>previous cycle), is given in Eq. (3.1.17b):</p><p>images</p><p>where p is the proportion of cars turning left. This result could be</p><p>evaluated as it stands for any values of p and v, but it is more informative to</p><p>reason to another, simpler form.</p><p>If the probability is p that any particular car will turn left, then we</p><p>can consider directly the arrival only of those cars which desire to turn left.</p><p>Returning to the derivation of the Poisson distribution from the binomial,</p><p>one can see that, since such left-turning cars arrive with a probability † p</p><p>times the probability that any car arrives, the number X of these left-turning</p><p>cars is also Poisson-distributed, but with parameter pv.‡ Hence the</p><p>probability that the left-turn lane is inadequate is simply the probability that</p><p>X is greater than or equal to four, or§:</p><p>images</p><p>For ν = 6 and p = 0.3, pν = 1.8 and</p><p>images</p><p>If this number is considered by the engineer to be too large, he might</p><p>consider increasing the capacity of the lane in order to reduce the likelihood</p><p>of inadequate performance.</p><p>It is important to realize the generality of the result used here. The</p><p>implication is that if a random variable Z is Poisson-distributed, then so too</p><p>is the random variable X, which is derived by (independently) selecting only</p><p>with probability p each of the incidents counted by Z; that is if Z is P(ν),</p><p>then X is P(pν). More formally, the distribution of X is found as</p><p>images</p><p>The conditional term is simply the probability of observing x successes</p><p>in z Bernoulli trials; thus</p><p>images</p><p>which, upon changing the variable to u = z – x, reduces to</p><p>images</p><p>Examples of application of this result might include X being the</p><p>number of hurricane-caused floods greater than critical magnitude when Z is</p><p>the total number of hurricane arrivals, or X being the number of vehicles</p><p>recorded by a defective counting device when Z is the total number of</p><p>vehicles passing in a given interval. The latter example was encountered in</p><p>Sec. 2.2.2 when the joint distribution of X and Z was investigated.</p><p>3.2.2 Time Between Events: The Exponential Distribution</p><p>The traffic engineer observing a traffic stream is often concerned with the</p><p>length of the time interval between vehicle arrivals at a point. If an interval</p><p>is too short, for example, it will cause a car attempting to cross or merge</p><p>with the traffic</p><p>anchor force.</p><p>images</p><p>Compute the sample mean and variance of each set of data. Construct</p><p>histograms, frequency distributions, and cumulative frequency</p><p>distributions. Assuming that the jack forces are constant, what can be said</p><p>about the influence of nominal curvature on friction loss?</p><p>1.4. The following values of shear strength (in tons per square foot) were</p><p>determined by unconfined compression tests of soil from Waukell Creek,</p><p>California. Plot histograms of this data using four, six, and eight intervals.</p><p>In the second case consider three different locations of the interval division</p><p>points. With a few extreme points, as here, it may be advantageous to</p><p>consider unequal intervals, longer in the right-hand tail.</p><p>images</p><p>1.5. Compute the individual sample means, standard deviations, and</p><p>coefficients of variation for the basement, fourth floor, and ninth floor of</p><p>the floor-load data in Table 1.2.1. How do they compare with each other?</p><p>Why might such differences arise?</p><p>1.6. The following data for the Ogden Valley artesian aquifer have been</p><p>collected over a period of years. Find the sample means, variances,</p><p>standard deviations, and correlation coefficient.</p><p>images</p><p>1.7. The maximum annual flood flows for the feather River at Oroville,</p><p>California, for the period 1902 to 1960 are as follows. The data have been</p><p>ordered, but the years of occurrence are also given.</p><p>images</p><p>images</p><p>Compute sample mean and variance. Plot histogram and frequency</p><p>distribution. If a 1-year construction project is being planned and a flow of</p><p>20,000 cfs or greater will halt construction, what, in the past, has been the</p><p>relative frequency of such flows?</p><p>1.8. The piezometer gauge pressures at a critical location on a penstock</p><p>model were measured under maximum flow conditions as follows (ordered</p><p>data in inches of mercury).</p><p>images</p><p>Compute sample mean, variance, and skewness coefficient. Use a</p><p>histogram with the mean at an interval boundary to answer the question:</p><p>did values above and below the mean occur with equal frequency? Were</p><p>very low readings as common as very high readings?</p><p>1.9. Embankment material for zone 1 of the Santa Rosita Dam in Southern</p><p>Chihuahua, Mexico, will come from a borrow pit downstream from the</p><p>dam site at a location that is frequently flooded. A cofferdam 800 m long is</p><p>needed and the contractor needs to know the optimum construction height.</p><p>Normal flow (200 m3/sec) requires a height of 3 m. Flooding will involve a</p><p>3-week delay in construction. Maximum flow rates from 1921 to 1965</p><p>were:</p><p>images</p><p>The contractor’s options are:</p><p>images</p><p>The cost of a 3-week delay from flooding of the borrow pit is estimated as</p><p>$30,000.</p><p>Compute the sample mean and variance. Will a histogram be useful in the</p><p>analysis of the contractor’s decision? Why? How would you structure the</p><p>decision situation? How does time enter the problem?</p><p>1.10. In heavy construction operations the variation in individual vehicle</p><p>cycle times can cause delays in the total process. Find the sample means,</p><p>standard deviations, and coefficients of variation for each of the following</p><p>specific steps in a cycle and for the total cycle time. Which steps are “most</p><p>variable?” Which contribute most seriously to the variations in the total</p><p>cycle time? Is there an indication that some vehicle/driver combinations are</p><p>faster than others? (Check by calculated haul and return time correlation.)</p><p>How does the standard deviation of the sum compare to those of the parts?</p><p>How does the coefficient of variation of the sum compare to those of the</p><p>parts?</p><p>images</p><p>1.11. Total cycle times of trucks hauling asphaltic concrete on a highway</p><p>project were observed and found to be (in minutes):</p><p>images</p><p>Find the sample mean, standard deviation, skewness coefficient, and</p><p>coefficient of kurtosis of this set of data. Plot its histogram.</p><p>1.12. Fifteen lots of 100 sections each of 108-in. concrete pipe were tested</p><p>for porosity. The number of sections in each lot failing to meet the</p><p>standards were:</p><p>images</p><p>Compute the sample mean, variance, and coefficient of variation.</p><p>If the plant continues to manufacture pipe of this quality, can you suggest a</p><p>possible technique for quality control of the product? What cost factors</p><p>enter the problem?</p><p>1.13. The times (in seconds) for loading, swinging, dumping, and returning</p><p>for a shovel moving sandstone on a dam project were measured as shown</p><p>in table at top of p. 24.</p><p>Compute sample mean, variance, and coefficient of variation of each set of</p><p>data. If the variability in total time is causing hauling problems, which</p><p>operation should be studied as the primary source of variability in the total?</p><p>Which of the summary statistics would be most useful in such a study?</p><p>images</p><p>1.14. The San Francisco–Oakland Bay Bridge provides an interesting</p><p>opportunity for the study of highway capacity as a random phenomenon. A</p><p>device located at the toll booth summarizes traffic flow by direction in 6-</p><p>min increments. Maximum flow each day has been recorded as follows for</p><p>a survey duration of 2 days.</p><p>images</p><p>(a) Check for growth trend.</p><p>(b) Plot the histogram and compute the numerical summaries using all 14</p><p>data points.</p><p>(c) What trends will become apparent if the bridge is approaching its</p><p>maximum daily capacity? Consider the influence of variation in demand</p><p>associated with users’ knowledge of congestion periods.</p><p>1.15. The traffic on a two-lane road has been studied in preparation for the</p><p>start of a construction project during which one lane will be closed. The</p><p>problem is whether to use an automatic signal with a fixed cycle, a</p><p>flagman, or go to the expense of constructing a new temporary lane. The</p><p>data are:</p><p>images</p><p>Compute sample mean and variance of each set. What can you say about</p><p>the traffic pattern that tends to discriminate between an automatic signal</p><p>with a fixed cycle and a flagman who can follow the traffic demand? How</p><p>would you make a study of the possible demand for a new temporary lane?</p><p>Hint: How does the detour travel time influence the problem? What data</p><p>are needed?</p><p>1.16. Soil resistivity is used in studies of corrosion of pipes buried in the</p><p>soil. For example, a resistivity of 0 to 400 ohms/cm represents extremely</p><p>severe corrosion conditions; 400 to 900, very severe; 900 to 1500, severe;</p><p>1500 to 3500, moderate; 3500 to 8000, mild; and 8000 to 20,000, slight</p><p>risk. There were 32 measurements of soil resistivity made at a prospective</p><p>construction site.</p><p>images</p><p>Compute sample mean and variance. Construct a histogram using the</p><p>classification bands above as intervals. Illustrate the same relations on the</p><p>cumulative frequency distribution and indicate the observed relative</p><p>frequency of occurrence. Note that the mean represents a spatial mean and</p><p>that the variance is a measure of lack of homogeneity of the resistivity</p><p>property over the site.</p><p>1.17. The water-treatment plant at an air station in California was</p><p>constructed for a design capacity of 4,500,000 gal/day (domestic use). It is</p><p>nearly always necessary to suspend lawn irrigation when demand exceeds</p><p>supply. There are, of course, attendant losses. Measured demands during</p><p>July and August 1965 (weekdays only) were (in thousands of gallons per</p><p>day, ordered data):</p><p>images</p><p>Compute sample mean and variance. Construct a cumulative histogram in</p><p>which 4,500,000 gal/day is one of the interval boundaries. On a relative</p><p>frequency basis, how often did demand exceed capacity?</p><p>1.18. The Stanford pilot plant attempts to treat the effluent of an activated</p><p>sludge sewage-treatment plant alternating between a diatomite filter and a</p><p>sand filter.</p><p>images</p><p>The diatomaceous earth filter fouls quickly if the raw water contains too</p><p>many suspended solids. The sand filter is then used during periods in which</p><p>the diatomite filter is inoperative. If x is the time of operation of the</p><p>diatomite filter and y the operating time of the sand filter during cleaning of</p><p>the diatomite filter, the data are a typical sample of paired observations.</p><p>Compute the sample mean and variance of each set and the correlation</p><p>coefficient. Plot</p><p>a scatter diagram. Does the plot verify the calculated</p><p>correlation coefficient?</p><p>1.19. Show that the following alternate, more easily computed form of the</p><p>sample covariance is equivalent to Eq. (1.3.1).</p><p>images</p><p>1.20. The percentage frequencies of wind direction and speed on an annual</p><p>basis for a 10-year study at the Logan Airport, Boston, Massachusetts, are:</p><p>Hourly observations of wind speed, mph*</p><p>images</p><p>* + indicates frequency less than 1 percent.</p><p>(a) Make a scatter diagram for direction versus average speed, one point for</p><p>each 1 percent. Make a similar study for direction versus 8 to 12 and versus</p><p>13 to 18 mph wind speeds. Why is the first diagram different from the other</p><p>two?</p><p>(b) Compute the sample mean and sample variance of the average speed.</p><p>(c) Compute the mean and variance of direction (in degrees from N). What</p><p>should you do with the two percent of calm days?</p><p>1.21. One procedure for setting the allowable speed limit on a highway is to</p><p>assume that the proper limit has a 15 percent chance of being exceeded</p><p>based on a spot speed study. If this decision rule is used, what speed limit</p><p>do you recommend for U.S. Route 50 and Main Street? No limit existed</p><p>when these data were observed:</p><p>U.S. Route 50</p><p>images</p><p>Main street</p><p>images</p><p>(a) Plot the histograms for speed of passenger cars, trucks, and all vehicles</p><p>on U.S. Route 50 and on Main Street. On U.S. Route 50, is a dual speed</p><p>limit for passenger cars and trucks reasonable?</p><p>(b) Compute the mean, variance, and coefficient of variation of speed for</p><p>all vehicles on U.S. Route 50 and for those on Main Street.</p><p>(c) If you were presenting your study to the City Council, what would you</p><p>recommend as the speed limits? Why? If the council is very sensitive to</p><p>local political pressure, would your recommendation change, assuming that</p><p>the quoted decision rule is only a rough guide? The Main Street traffic is</p><p>primarily local traffic.</p><p>1.22. Show that in situations such as that in Prob. 1.10, where interest</p><p>centers on both the parts and their sum, the sample variance sT</p><p>2 of the sum</p><p>of two variables equals:</p><p>images</p><p>where s1</p><p>2, s2</p><p>2, and r12 are, respectively, the sample variances and sample</p><p>correlation coefficient of variables 1 and 2. Show, too, that:</p><p>images</p><p>In general, for a number n of variables and their sum,</p><p>images</p><p>Notice that if the rij’s are small in absolute value,</p><p>images</p><p>But in general sT</p><p>2 may be larger or smaller than this sum. See Prob. 1.10.</p><p>1.23. One of the factors considered when determining the susceptibility to</p><p>liquefaction of sands below a foundation during earthquake motions is their</p><p>penetration resistance as measured by blow counts (number of blows per</p><p>foot of penetration). These measurements show marked dispersion,</p><p>especially in soils whose liquefaction susceptibility might be in question.</p><p>Therefore histograms of blow-count tests made at different depths and</p><p>locations in the soil layer are compared in shape with histograms of blow</p><p>counts at sites where liquefaction has or has not occurred during</p><p>earthquakes. At a site under investigation the following data are made</p><p>available:</p><p>images</p><p>(a) Prepare a histogram of this data. Use three different vertical axes with</p><p>scales: “Number of tests per 10-blow interval,” “frequency of tests per 10-</p><p>blow interval,” and “frequency of tests per 1-blow interval.”</p><p>(b) The following histograms are presented as shown in a report on the</p><p>large 1964 Niigata Earthquake in Japan. As is all too frequently the case in</p><p>engineering reports, it is not clear how the histograms were constructed.</p><p>Assume that the vertical axis is “frequency per five-blow interval” and that</p><p>the dots represent the midpoints of the intervals –2.4 to +2.5, 2.6 to 7.5, 7.6</p><p>to 12.5, etc. In order to aid in judging the susceptibility to liquefaction of</p><p>the present site, replot these two histograms on the histogram prepared in</p><p>part (a) above. Which of the three vertical scales should be used to avoid</p><p>most easily the difficulties caused by the different numbers of tests and</p><p>different interval lengths?</p><p>(c) Instead of comparing entire histograms, what sample averages might</p><p>one compute for the various sets of data to aid in this judgment? Do you</p><p>think the average is sufficient? Is the skewness coefficient helpful?</p><p>images</p><p>Fig. P1.23 Histograms of data from Niigata earthquake.</p><p>1.24. In a study of rainstorms, the following data were observed at a</p><p>number of recording stations.</p><p>images</p><p>Source: E. L. Grant [1938], Rainfall Intensities and Frequencies, ASCE</p><p>Trans., vol. 103, pp. 384–388.</p><p>(a) Plot a histogram and a cumulative frequency polygon for this data.</p><p>Compute the sample mean and sample standard deviation.</p><p>(b) How do these data differ in character from stream flow data?</p><p>† When constructing the cumulative frequency distribution, one can avoid</p><p>the arbitrariness of the intervals by plotting one point per observation, that</p><p>is, by plotting i/n versus x(i), where x(i) is the ith in ordered list of data (see</p><p>Fig. 1.2.1).</p><p>† If advantageous, unequal interval widths may be preferable (see Sec. 4.4).</p><p>† The reader would do well to plot a number of different histograms of this</p><p>failure-load data, both to judge the value of this means for data</p><p>interpretation in this example and to illustrate the remarks about histogram</p><p>intervals made in Sec. 1.1.</p><p>† A fourth numerical summary, the coefficient of kurtosis, may also be</p><p>employed. Without large sample sizes, however, its use is seldom</p><p>recommended.</p><p>The sample coefficient of kurtosis g2 is related to the “peakedness” of</p><p>the histogram:</p><p>images</p><p>Traditionally the value of this coefficient is compared to a value of g2 =</p><p>3.0, the kurtosis coefficient of a commonly encountered bell-shaped</p><p>continuous curve which we shall learn to call the normal curve (Sec. 3.3.3).</p><p>For the data here,</p><p>images</p><p>2</p><p>Elements of Probability Theory</p><p>Every engineering problem involves phenomena which exhibit scatter of the</p><p>type illustrated in the previous chapter. To deal with such situations in a</p><p>manner which incorporates this variability in his analyses, the engineer</p><p>makes use of the theory of probability, a branch of mathematics dealing with</p><p>uncertainty.</p><p>A fundamental step in any engineering investigation is the formulation of a</p><p>set of mathematical models—that is, descriptions of real situations in a</p><p>simplified, idealized form suitable for computation. In civil engineering, one</p><p>frequently ignores friction, assumes rigid bodies, or adopts an ideal fluid to</p><p>arrive at relatively simple mathematical models, which are amenable to</p><p>analysis by arithmetic or calculus. Frequently these models are</p><p>deterministic: a single number describes each independent variable, and a</p><p>formula (a model) predicts a specific value for the dependent variable. When</p><p>the element of uncertainty, owing to natural variation or incomplete</p><p>professional knowledge, is to be considered explicitly, the models derived</p><p>are probabilistic and subject to analysis by the rules of probability theory.</p><p>Here the values of the independent variables are not known with certainty,</p><p>and thus the variable related to them through the physical model cannot be</p><p>precisely predicted. In addition, the physical model may itself contain</p><p>elements of uncertainty. Many examples of both situations will follow.</p><p>This chapter will first formalize some intuitively satisfactory ideas about</p><p>events and relative likelihoods, introducing and defining a number of words</p><p>and several very useful notions. The latter part of the chapter is concerned</p><p>with the definition, the description, and the manipulation of the central</p><p>character in probability—the random variable.</p><p>2.1 RANDOM EVENTS</p><p>Uncertainty is introduced into engineering problems through the variation</p><p>inherent in nature, through man’s lack of understanding of all the causes and</p><p>effects in physical systems, and through lack of sufficient data. For example,</p><p>even with a long history of data, one cannot predict the maximum flood that</p><p>will occur in the next 10 years in a given area. This uncertainty is a product</p><p>of natural variation. Lacking</p><p>a full-depth hole, the depth of soil to rock at a</p><p>building site can only be estimated. This uncertainty is the result of</p><p>incomplete information. Thus both the depth to rock and the maximum flood</p><p>are uncertain, and both can be dealt with using the same theory.</p><p>As a result of uncertainties like those mentioned above, the future can never</p><p>be entirely predicted by the engineer. He must, rather, consider the</p><p>possibility of the occurrence of particular events and then determine the</p><p>likelihood of their occurrence. This section deals with the logical treatment</p><p>of uncertain events through probability theory and the application to civil</p><p>engineering problems.</p><p>2.1.1 Sample Space and Events</p><p>Experiments, sample spaces, and events The theory of probability is</p><p>concerned formally with experiments and their outcomes, where the term</p><p>experiment is used in a most general sense. The collection of all possible</p><p>outcomes of an experiment is called its sample space. This space consists of</p><p>a set S of points called sample points, each of which is associated with one</p><p>and only one distinguishable outcome. The fineness to which one makes</p><p>these distinctions is a matter of judgment and depends in practice upon the</p><p>use to which the model will be put.</p><p>As an example, suppose that a traffic engineer goes to a particular street</p><p>intersection exactly at noon each weekday and waits until the traffic signal</p><p>there has gone through one cycle. The engineer records the number of</p><p>southbound vehicles which had to come to a complete stop before their light</p><p>turned green. If a minimum vehicle length is 15 ft and the block is 300 ft</p><p>long, the maximum possible number of cars in the queue is 20. If only the</p><p>total number of vehicles is of interest, the sample space for this experiment</p><p>is a set of 21 points labeled, say, E0, E1, . . ., E20, each associated with a</p><p>particular number of observed vehicles. These might be represented as in</p><p>Fig. 2.1.1. If the engineer needed other information, he might make a finer</p><p>distinction, differentiating between trucks and automobiles and recording the</p><p>number of each stopped. The sample space for the experiment would then be</p><p>larger, containing an individual sample point Ei,j for each possible</p><p>combination of i cars and j trucks such that the maximum value of i+j = 20,</p><p>as in Fig. 2.1.2.</p><p>images</p><p>Fig. 2.1.1 An elementary sample space—Ej implies j vehicles observed.</p><p>An event A is a collection of sample points in the sample space S of an</p><p>experiment. Traditionally, events are labeled by letters. If the distinction</p><p>should be necessary, a simple event is an event consisting of a single sample</p><p>point, and a compound event is made up of two or more sample points or</p><p>elementary outcomes of the experiment. The complement A c of an event A</p><p>consists of all sample points in the sample space of the experiment not</p><p>included in the event. Therefore, the complement of an event is also an</p><p>event.†</p><p>images</p><p>Fig. 2.1.2 Elementary sample space for cars and trucks. Eij implies i cars</p><p>and j trucks with a maximum of 20 including both types of vehicles.</p><p>In the experiment which involved counting all vehicles without regard for</p><p>type, the observation of “no stopped vehicles” is a simple event A, and the</p><p>finding of “more than 10 stopped vehicles” is a compound event B. The</p><p>complement of the latter is the event Bc, “10 or fewer stopped vehicles were</p><p>observed.” Events defined on a sample space need not be exclusive; notice</p><p>that events A and Bc both contain the sample point E0.</p><p>In testing the ultimate strength of reinforced-concrete beams described in</p><p>Sec. 1.2, the load values were read to the nearest 50 lb. The sample space for</p><p>the experiment consists of a set of points, each associated with an outcome,</p><p>0, 50, 100, . . ., or M lb, where M is some indefinitely large number, say,</p><p>infinity. A set of events of interest might be A0, A1, A2, ..., defined such that</p><p>A0 contains the sample points associated with loads 0 to 950 lb, A1 contains</p><p>those associated with 1000 to 1950 lb, and so forth.</p><p>In many physical situations, such as this beam-strength experiment, it is</p><p>more natural and convenient to define a continuous sample space. Thus, as</p><p>the measuring instrument used becomes more and more precise, it is</p><p>reasonable to assume that any number greater than zero, not just certain</p><p>discrete points, will be included among the possible outcomes and hence</p><p>defined as a sample point. The sample space becomes the real line, 0 to ∞.</p><p>In other situations, a finite interval is defined as the sample space. For</p><p>example, when wind directions at an airport site are being observed, the</p><p>interval 0 to 360° becomes the sample space. In still other situations, when</p><p>measurement errors are of interest, for example, the line from – ∞ to + ∞ is a</p><p>logical choice for the sample space. To include an event such as ∞ in the</p><p>sample space does not necessarily mean the engineer thinks it is a possible</p><p>outcome of the experiment. The choice of ∞ as an upper limit is simply a</p><p>convenience; it avoids choosing a specific, arbitrarily large number to limit</p><p>the sample space.</p><p>Events of interest in the beam experiment might be described as follows‡</p><p>(when the sample space is defined as continuous):</p><p>images</p><p>images</p><p>Fig. 2.1.3 Continuous sample space for beam-failure load.</p><p>D1 is a simple event; D2 and D3 are compound events. They are shown</p><p>graphically in the sample space 0 to ∞ in Fig. 2.1.3.</p><p>Relationships among events Events in a sample space may be related in a</p><p>number of ways. Most important, if two events contain no sample points in</p><p>common, the events are said to be mutually exclusive or disjoint. Two</p><p>mutually exclusive events—A, “fewer than 6 stopped vehicles were</p><p>observed” and B, “more than 10 stopped vehicles were observed”—are</p><p>shown shaded in the sample space of the first vehicle-counting experiment</p><p>(Fig. 2.1.4). The events D1 and D3 defined above are mutually exclusive. D2</p><p>and D3 are also mutually exclusive, owing to the care with which the</p><p>inequality (≤) and strict inequality (>) have been written at 10,000.</p><p>The notion of mutually exclusive events extends in an obvious way to more</p><p>than two events. By their definition simple events are mutually exclusive.</p><p>If a pair of events A and B are not mutually exclusive, the set of points</p><p>which they have in common is called their intersection, denoted A ∩ B. The</p><p>intersection of the event A defined in the last paragraph and the event C,</p><p>“from four to eight stopped vehicles were observed,” is illustrated in Fig.</p><p>2.1.5. The intersection of the events D1 and D2 in Fig. 2.1.3 is simply the</p><p>event D1 itself. If the intersection of two events is equivalent to one of the</p><p>events, that event is said to be contained in the other. This is written D1 ⊂</p><p>D2.</p><p>images</p><p>Fig. 2.1.4 Mutually exclusive events. Compound events A and B are</p><p>mutually exclusive; they have no sample points in common.</p><p>images</p><p>Fig. 2.1.5 Intersection A ∩ C, comprising events E4 and E5, is the</p><p>intersection of compound events A and C.</p><p>The union of two events A and C is the event which is the collection of all</p><p>sample points which occur at least once in either A or C. In Fig. 2.1.5 the</p><p>union of the events A and C, written A images C, is the event “less than</p><p>nine stopped vehicles were observed.” The union of the events D2 and D3 in</p><p>Fig. 2.1.3 is the event that the failure load is greater than 9000 lb. The union</p><p>of D1 and D2 is simply D2 itself.</p><p>Two-dimensional and conditional sample spaces For purposes of</p><p>visualization of certain later developments, two other types of sample spaces</p><p>deserve mention. The first is the two- (or higher) dimensional sample space.</p><p>The sample space in Fig. 2.1.2 is one such, where the experiment involves</p><p>observing two numbers, the number of cars and the number of trucks. It</p><p>might be replotted as shown in Fig. 2.1.6; there each point on the grid</p><p>represents a possible outcome, a sample point. In the determination of the</p><p>best orientation of an airport runway, an experiment might involve</p><p>measuring both wind speed and direction. The continuous sample space</p><p>would appear</p><p>as in Fig. 2.1.7, limited in one dimension and unlimited in the</p><p>other. Any point in the area is a sample point. An experiment involving the</p><p>measurement of number and average speed of vehicles on a bridge would</p><p>lead to a discrete-continuous two-dimensional sample space.</p><p>The second additional kind of sample space of interest here is a conditional</p><p>sample space. If the engineer is interested in the possible outcomes of an</p><p>experiment given that some event A has occurred, the set of events</p><p>associated with event A can be considered a new, reduced sample space. For,</p><p>conditional on the occurrence of event A, only the simple events associated</p><p>with the sample points in that reduced space are possible outcomes of the</p><p>experiment. For example, given that exactly one truck was observed, the</p><p>conditional sample space in the traffic-light experiment becomes the set of</p><p>events E0,1,El,1, . . ., E19,1. Given that two or fewer trucks were observed,</p><p>the conditional sample space is that illustrated in Fig. 2.1.8. Similarly, the</p><p>airport engineer might be interested only in higher-velocity winds and hence</p><p>restrict his attention to the conditional sample space associated with winds</p><p>greater than 20 mph, leading to the space shown in Fig. 2.1.9. Whether a</p><p>sample space is the primary or the conditional one is clearly often a matter</p><p>of the engineer’s definition and convenience, but the notion of the</p><p>conditional sample space will prove helpful.</p><p>images</p><p>Fig. 2.1.6 Discrete two-dimensional sample space. Each sample point</p><p>represents an observable combination of cars and trucks such that their sum</p><p>is not greater than 20.</p><p>images</p><p>Fig. 2.1.7 Continuous two-dimensional sample space. All possible</p><p>observable wind speeds and directions at an airport are described.</p><p>images</p><p>Fig. 2.1.8 Discrete conditional sample space. Given that two or fewer trucks</p><p>were observed (see Fig. 2.1.6).</p><p>images</p><p>Fig. 2.1.9 Continuous conditional sample space. Describes all possible wind</p><p>directions and velocities given that the velocity is greater than 20 mph.</p><p>For the remainder of Sec. 2.1 we shall restrict our attention to one-</p><p>dimensional discrete sample spaces, and shall return to the other important</p><p>cases only after the introduction in Sec. 2.2 of the random variable.</p><p>2.1.2 Probability Measure</p><p>Interpretation of probabilities To each sample point in the sample space of</p><p>an experiment we are going to assign a number called a probability measure.</p><p>The mathematical theory of probability is not concerned with where these</p><p>numbers came from or what they mean; it only tells us how to use them in a</p><p>consistent manner. The engineer who puts probability to work on his models</p><p>of real situations must be absolutely sure what the set of numbers he assigns</p><p>means, for the results of a probabilistic analysis of an engineering problem</p><p>can be helpful only if this input is meaningful.</p><p>An intuitively satisfying explanation of the probability measure assigned to</p><p>a sample point is that of relative frequencies. If the engineer assigns a</p><p>probability measure of p to a sample point in a sample space, he is usually</p><p>willing to say that if he could make repeated trials of the same experiment</p><p>over and over again, say M times, and count the number of times N that the</p><p>simple event associated with this sample point was observed, the ratio of N</p><p>to M would be very nearly p. One frequently hears, for example, that the</p><p>probability of a tossed coin coming up “heads” is one-half. Experience has</p><p>shown that this is very nearly the fraction of any series of a large number of</p><p>tosses of a well-balanced coin that will show a head rather than a tail. This</p><p>interpretation of the probability measure is commonly adopted in the</p><p>physical sciences. It is considered quite objectively as a property of certain</p><p>repetitious phenomena. When formalized through limit theorems, the notion</p><p>of relative frequency can serve as a basis as well as an interpretation of</p><p>probability (Von Mises [1957]). Relative frequency, when it applies, is</p><p>without question a meaningful and useful interpretation of the probability</p><p>measure.</p><p>But what is the engineer to do in a situation where such repetition of the</p><p>experiment is impossible and meaningless? How does one interpret, for</p><p>instance, the statement that the probability is 0.25 that the soil 30 ft below a</p><p>proposed bridge footing is not sand but clay? The soil is not going to be clay</p><p>on 1 out of every 4 days that it is observed; it is either clay or it is not. The</p><p>experiment here is related to an unknown condition of nature which later</p><p>may be directly observed and determined once and for all.</p><p>The proved usefulness in bringing probability theory to bear in such</p><p>situations has necessitated a more liberal interpretation of the expression</p><p>“the probability of event A is p.” The probabilities assigned by an engineer</p><p>to the possible outcomes of an experiment can also be thought of as a set of</p><p>weights which expresses that individual’s measure of the relative likelihoods</p><p>of the outcomes. That is, the probability of an event might be simply a</p><p>subjective measure of the degree of belief an engineer has in a judgment or</p><p>prediction. Colloquially this notion is often expressed as “the odds are 1 to 3</p><p>that the soil is clay.” Notice that if repetitions are involved, the notions of</p><p>relative frequencies and degree of belief should be compatible to a</p><p>reasonable man. Much more will be said of this “subjective” probability in</p><p>Chap. 5, which includes methods for aiding the engineer in assessing the</p><p>numerical values of the probabilities associated with his judgements. This</p><p>interpretation of probability, as an intellectual concept rather than as a</p><p>physical property, also can serve as a basis for probability theory.</p><p>Engineering students will find Tribus’ recent presentation of this position</p><p>very appealing (Tribus [1969]).</p><p>Like its interpretations, the sources of the probability measure to be assigned</p><p>to the sample points are also varied. The values may actually be the results</p><p>of frequent observations. After observing the vehicles at the intersection</p><p>every weekday for a year, the traffic engineer in the example in the previous</p><p>section might assign the observed relative frequencies of the simple events,</p><p>“no cars,” “one stopped car,” etc., to the sample points E0, E1 . . ., E20.</p><p>Reflecting the second interpretation of probability, the probability measure</p><p>may be assigned by the engineer in a wholly subjective manner. Calling</p><p>upon past experience in similar situations, a knowledge of local geology, and</p><p>the taste of a handful of the material on the surface, a soils engineer might</p><p>state the odds that each of several types of soil might be found below a</p><p>particular footing.</p><p>Finally, we shall see that through the theory of probability one can derive the</p><p>probability measure for many experiments of prime interest, starting with</p><p>assumptions about the physical mechanism generating the observed events.</p><p>For example, by making certain plausible assumptions about the behavior of</p><p>vehicles and knowing something about the average flow rate, that is, by</p><p>modeling the underlying mechanism, the engineer may be able to calculate a</p><p>probability measure for each sample point in the intersection experiment</p><p>without ever making an actual observation of the particular intersection.</p><p>Such observation is impossible, for example, if the intersection is not yet</p><p>existent, but only under design. As in deterministic problem formulations,</p><p>subsequent observations may or may not agree with the predictions of the</p><p>hypothesized mathematical model. In this manner, models (or theories) are</p><p>confirmed or rejected.</p><p>Axioms of probability No matter how the engineer chooses to interpret the</p><p>meaning of the probability measure and no matter what its source, as long as</p><p>the assignment of these weights is consistent with three simple axioms, the</p><p>mathematical validity of any results derived through the correct application</p><p>of the axiomatic theory of probability is assured. We use the notation† P[A]</p><p>to denote the probability of an event A, which in the context</p>