Probability Concepts in Engineering 2nd Ed,Item Preview
WebSep 24, · The second edition of this well-known book (previously titled Probability Concepts in Engineering Planning and Design) by Alfredo Ang and Wilson Tang, two WebOct 25, · Probability concepts in engineering V1 2 ed - Ang y blogger.com - Free ebook download as PDF File .pdf) or read book online for free. Scribd is the world's largest WebProbability Concepts in Engineering Emphasis on Applications in Civil & Environmental Engineering 2NDEDITION Prob ab iIity Concepts in Engineering* Emphasis on WebProbability Concepts in Engineering: Emphasis on Applications to Civil and Environmental Engineering [2 ed.] , X. Apply the principles WebProbability concepts in engineering planning and design by Ang, Alfredo Hua-Sing, Publication date Openlibrary_edition OLM Openlibrary_work ... read more
conditions; and available earthquake Tecords , the annual mean occurrence rate of earthquakes with damaging intensity at. site A is estimated to be O. Ql or 0. The occurrence of an earthquake may be modeled by a Poisson process. Assume that earthquake damages to the dam will be repaired after eacfi earthquake and the dam restored to its original condition. Let X be the number of earthquakes occurring in the life of the dam, assumed to be 50 years. All losses are measured in terms of million dollars. LE[LIB, ·,·. o21 0. fi56 million Case3 Finally, if the earthquake damage function follows an exponentia1loss function,.
ance cost through the years. Table E2. Depending on the specific loss function, A orB may be the optimal site for the dam. Because of bad weatber in the first phase of the project, construction has not been proceeding at the anticipated rate. Based on the contractor's judgment, the project could still meet the deadline if exceptionally good weather condition G prevails in the next 30 days. However, ifnormal weather N prevails. completion will be delayed With. a mean delay of 5 days and a standard deviation of 3 days beyond the scheduled completion date; if bad weather B continues. the corresponding mean and standard deviation of delay will be 15 and 5 days,-respcctively. Assume that normal and bad weather conditions are equally likely to occur in the remaining time required and the cost of delay is 0.
This cost function implies that the loss per day of delay increases as the delay becomes longer. The contractor has an option to launch a crash progr. With this crash program. if a delay does occur. its mean and standard deviation will both be: reduced to I dav under normal weather: and 5 and 2 days, respectively, under bad weather. Solution From the decision tree shown in Fig. Ix 2 Figure E2. According to this analysis, the contractor should launch a crash program to complete the construction project on-schedule. With instrument. I , the surveyor must measure 10 separate segments. The error in each segment is random with mean zero and standard deviation u1 ; u 1 is estimated to have a mean oft inch and c. segments to measure the distance AB; the error in each segment has mean zero and standard deviation with a mean of3 inches and c.
The cost of j:lerforming. Which of the instruments should thf' surveyor usc based on the ~aximum EMV criterion? f}nu f a do- 2. THE DECISION MODEL 23 EXAMPLE 2. power supply; m:reation. and ftood control. After a detailed economic analysis. the cost of the damage as a function of the discharge le,·el of the river is tabulated in Table E2lla. Determine the expected annual cost of damage. Table El. lla Damage Cost at Given Discharge Level Discharge I cfs · · Cost of Damage in S! OOO 60 0 80 1 I ! Figure £2. From tbe flow fre- quency curve of Fig. In the next part of this eximiple see below. other alternatives for improving the spillway ·capacity are considered.
The expected 11Jlnual damage cost for each alternative is also computed using the same technique. llb Summary of Computations of Expected Annual Risk Cost Average Dis·. To improve the capacity. three alternative designs are considered: a1 : Lengthening the spillway. aJ: Lengthening the spillway, considerable crest lowering. and installing radial gates. The capital costs required in the three alternatives arc S 1. OOO, respectively. which is 0. llc compares the annual costs for the various alternatives. Ti c existing design is as59 :iated with. the minimum total annual cost; hence, no improvement should be implemented on the existing structure.
HODEL 25 ° Table EZ. llc Summary of Annual Costs in Dollars Annual Average Annual Total Annual Total Equivalent Operation and Expected Annual Alternatives Capital Cost Maintenance Cost Risk Cost Cost uo 0 0 oo0 0 it is common in engineering to conduct research studies or perform laboratory or field tests before a final decision is made. This occurs especially when a decision involves unusual and innovative design concepts. or when time is available for collecting additional information. The acquisition of additional · information. energy, and financial resources; in spite of these added. the additionaiinformation may be valuable and will generally improve the chance of making a better decision. The added cost for this new information should be included or reflected in the decision analysis.
In general. the additional information would not eliminate all the uncertainties in a decision problem. At best. based on the experimental outcome, the probabilities in the decision tree would be updated by applying Bayes' Theorem see Chapters 2 and 8, Vol. A decision analysis with additional information commonly ref~:rred to as terminal analysis is similar to the prior analysis, except that the updated probabilities probabilities conditional on the experimental outcome are used in the computation of the expected monetary value or utilily. The following examples illustrate the terminal decision analysis. to adopt the smaller design, that is, unit A; whereas choosing the conversative design. unit B, would mean a large initial inv. After conferring with·his client, the engineer decided to find out more about the proposed waste treatment process before adopting a specific design.
The engineer ordered the construction of a laboratory model to simulate the proposed waste treatment process. The cost of constructing the experiment is SIO,OOO, and the reliability of the experimental 'results is as follows: if tbe efficiency of tbe proposed treatment process is actually high EH , the probability that the experimental results will indicate a high efficiency rating is 0. On the other hand, if tbe efficiency of the process is actually low, the probabilities that the experimental results will show high, medium,. and low efficiency ratings are 0. In short, the conditional probabilities are as follows:. P HRIEJi ';.. and low efficiency ratings, respectively. B El 0. Suppose ~be experiment·was performed and the results indicated a high rating. In such a case, the updl!. EL P' EL 0.
The corresponding updated probability of EL is 0. Ea, where the probabilities for. aU the paths arc revised in accordance with the HR test outcome. e expected monetary value of the two alternative desi!! design A should ·be chosen. if the test outcome indicates a medium rating MR , the updated probabilities of · E! f and EL arc, respectively, 0. The corresponding Rei at ive Monetary Value IS IOOOJ EH Ql A El Ql Test MR " EHIQl. B· EL HODEL 2i Relative Monetary Value ISIOOOl EH 0. revised expected monetary values for the two designs will be: £. In this case, the experimental evidence is not very useful; the test results fall in a fuz:zy region. course, the chance of this occurring may be reduced ifa more discriminating experiment is performed.
Finally, in Fig. if the test indicates a low rating LR , the updated probabilities of EH and EL will be0. ita were obtained revealing that over a year period. one earthquake of damaging intensity had occurred in site A, whereas no earthquakes were recorded at site B. Would the decision on the optimal dam site·be different given this new piece dr info·rmation? Assume that the average earthquake damage costs are 5X million dollars}, where X is the number of earthquake occurrences i. For site A. l ~ OOS P' v. a ""0. The expected seismic damage.. loss for each alternative is now computed based on these updated probabilities. the net expected benefits for site A and site B are, respectively. Therefore, with the additional information, site B becomes the preferred location for the dam. the analysis assumes that the additional information or. experimental outcome ls available. However, of equal or greater interest may be the question: ''Should additional information be obtained'r Of course, additional information, such as performing an experiment, involves the added cost of time and money: therefore, t~e answer to this question depends.
on Jhe benefit of the new infonna tion relative ~o the· add! Decisions involving whether additional information should. A two-stage decision problem is involved. At stage A, a decision on whether or not to proceed with an experiment is required; that is, a choice between an experiment a 3 and no. chosen, another decision at stage B is needed, depending on the particular experimental outcome ZJ. A sub- tree may be constructed for each experimental outcome zi at stage B, as shown in Fig. In a preposterior analysis. terminal analyses are first. performed on each of the subtrees as indicated in Fig. HODEL 19 No Experiment a3 Experiment Staqe A Figure 2. The maximum ·expected monetary value associ- ated with the optimal alternative of each subtree, namely dt, is then.
Comparing the expected utilities of a h a 1 with that' of a 3 experiment , the opt4rial alternative at stage A can be selected. If the alternative a 3 is selected, the decision maker must wait until a specific experimental outcome is obtained; at this point the corresponding optimal alternative as indicated by an asterisk is selected. The probabilities that the test will indicate HR. and LR are computed by applying Eq. For example,. In each of the subtrees, the optimal alternative is denoted by the asterisk and the corresponding optimal monetary value is shown;for example, the EMV for the subtree with outcome HR is namely - ~0 and ~41, ~:espectively from Example 2 7 , the engineer should opt for the experiment.
Then, depending on the test outcome. design A-orB may be selected accordingly, as indicated in Fig. Hence, if VI exceeds the experimental cost, the test' aitemative should be selected. The value of the additional information, however, is bounded by a limit referred to as the.. value of perfect information" VPI. Suppose a perfect test is available that will definitely tell whether the efficiency of the treatment process is high or low. The cor- responding subtree is shown in Fig. E; the cost of conducting the perfect test is not included. The events EH 0 , EL 0 represent the possible perfect test results; accordingly, the subsequent conditional probabilities of EH and EL are either one or zero.
For the specific test program studied in Example 2. the test should be performed. Observe that the value of this test is still considerably less thlJ. II the value of perfect information. Hence, more sophisticated test programs could be considered for reducing the uncertainties of the-efficiency of the treatment process, as long. Otherwise, the amount oT saving resulting from. ifst place. Errors in the estimated proba- bilities, therefore, are unavoidable. The effects of these errors on a decision are naturally of interest. In particular, the sensitivity of the optimal alternative to the values of the calculated probabilities is of interest. For instance, if the probability estimates are off by 10 'Y. will this alter the op. timal alternative? The following a example illustrates the nature of this problem and shows how sensitivity analysis may be performed. Instead of assigning 0. Pte corresponding probability of low efficiency is I - p.
Based on the decision tree ofFig. design B is the preferrc:d. On the other hand. design A would be preferred. based on the decision tree in Fig. a perfect test would be ofneg;ligible value. r·n indicate the.. ivcn value of p. lt attains the highest value at p ,. and any additional information can significantly increase the EMV of the decision. For the experiment proposed in Example 2. In generl! EMV of any imperfect test wili not be linear with p and it should lie between E PT and max EMV. that is. within the triangular shaded region shown in Fig.
The difference in the ordimtes of £ T and the max. EMV is VI. In short. if the number of alternatives is few. small errors in the ·estimated probability may not affect the selection of the optimal alternative, especially if the probabilitv is not close to the critical value sucli as 0. The sensitivity. analysis illustrated ab~ve can be extended to decision problems with more alternatives; howc;ver. in such. the 'arialysis w~uld naturally be more complex. HODEL 13 EMV ·-. A decision arialysis was performed to determine the possibility of mitigating the destructive force of hurricanes by seeding them with silver iodide before a hurricane hits the coast.
The decision tree for the study is shown in Fig. With the present seeding technology, the effect of seeding remains uncertain. Based on sub- jective: judgment and limited data from 'the seeding of Hurricane Debbie. It can be observed from these prob- abilities that wind speed tends to decrease with seeding; however, there is some probability that the hurricane wind speed may actuaUy increase after seeding. For this reason, the con- sequence of a seeding decision must include a ~government responsibility cost. he government will no~ be held responsible for damages, since people will accept hurricanes as natural.
On the other hand, if the government should order seeding and the hurricane wind ·speed actually grows stronger because· of it, the government may be subjectto lawsuits for hurricane-induced damages. These costs have been estimated and included in Fig. in the decision tree of Fig. Ea, the expected monetary value for the seeding and no seeding alternatives are. respectively, · ·. llt Total Ma1imum Damoqe R-sibility Cost Sustained Millions Cost l"-rtotit"' Millions Wind 5pHd o! of the potential liability cost to the t~ovemmcnt, because of the strong likelihood that property damages will be greatly reduced. A third alternative is to perform seeding experiments before deciding on whether or not to seed. Figure Eb shows the corresponding decision tree that includes the seeding experiments as an alternative. Corresponding to each experimental outcome, the probability of a particular change in wind speed in future hurricanes will be updated from the previously estimated values.
Representative paths or the decision tree a~e shown in Fig. b Suppose the performance of design B can be inveStigated with a mOdel test at a cost of S50, Should the test be performed? c At most how much should be spent to check if the assumptions are valid? the corresponding decision tree is given in Fig. Observe that the test result will only a1fcct the probabilities of whether the assumptions are valid or nilt, whereas the probability of satisfactory performance of the actual structure depending on the validity of the design assumptions are the same as those in part a. OI QI X A is the better design if the test model failed. The expected value of the test is E T "" 0. c The maximum cost of the test that can be justifiably spent may be determined by considering th~ perfect test.
The decision tree for this alternative is shown in Fig. Performing the ex~tecfloss calcllll! um amount that may be spent for checking the validity of the assumptions. To estimate the probability of lique- faction, the following simplified model is suggested. the stratum. ipre E2. where X is N , 50 and d is the depth in feet of the water table below the ground surface. Liquefaction occurs if S exceeds R. The present watenable ·is at the ground surface. However, the water table may be lowered by as much as 10 feet to impro~ the soil resistance against liqu~action. As the depth of the water table d increases, the resistance capacity of the sand against liquefaction increases, thereby reducing its risk of liquefaction during earthquakes; however, the associated cost oflowering the water table also increases rapidly.
The optimal value of a can be obtained by minimizing the sum of the cost oflowering the water table and the expected loss from liquefaction. as shown below, reveals that CT is minimum with d "" 4 ft. Therefore, if the water table is to be lowered, 4ft will be the optimal depth. b Upon further investigation, any lowering of the water table will not be permitted by a regulatory agency. However, the saturated sand can be excavated and replaced by material with a negligible chance ofliquefaction. The excavation will cost SO. S million. Should the soil be excavated or left untouched? The decision tree for this case is shown in Fig. Therefore, excavation is the preferred alternative, at a cost of SO. Determine the value of this test alternative. The distribution of W at the chance node A is N 50, obtained from the test. Ifw is less than. On the other hand, if w exceeds. l -w;40j2. he integral has been evaluated numerically. When we compare this with the expected monetary value of the optimal alternative without the test.
from part b. Three alternatives have been proposed by the Great Lakes Management to improve the situa- tion; namely: a1 : Build a new efficient treatment plant for SJO million. cost of SI. O million with 0. A founh alternative. Because of uncertainties in the environment, panic- ularly with respect to the potential sources of pollutants from industry, the level offuture water quality prior to treatment may be idealized as either q 1 or q2 , where q 1 is better 'than q1. five levels of improvement in water quality arc possible, namely I, II, III, IV, V, in decreasing degree of improvement. is dev. elopcd ~. in 4th year. Oix where the value of. For example, P II Ia 1q1 denotes the probability of achieving level II in water quality if alternative a 1 is chosen and the water quality prior to treatment is q 1 ; P llla 1qz.
T is the development time under alternativea2. x, and 0. lt can ·be observed from Fig. Eb · that each of the four alternatives could be the optimal one, depending on the value of x. For x less than 0. a~ will become the optimal alternative, implying that if water quality is extremely valuable, it would pay to build a new and efficient treatment plant even at a great cost. Because of frequent flood. damages to an adjacent town, a plan to increase the present levee elevatfon of 10 feet to either 14 or 16 feet is being investigated.
Based on the data collected from previous years, the annual flood level in the stream can be estimated quite accurately as having a median of 10 feet and a c. Suppose the true distribution could equally likely be normal or lognonnal. a Determine the optimal decision based on EMV criterion: b How much is it worth to verify the true distribution of the annual flood level? THE DECISION MODEL - 43 Figure E2. Solution a0 : Levee elevation remains at 10 feet. The alternatives are · a1: Increase levee to 14 feet. For each alternative the probability distribution of the annual maximum flood level, X, could ctthcr be lognormal LN or normal N ; on these bases, the probability ofin'!
In Millil! The total present cost of each alternative is the sum of the initial investment and the equivalent present value uf the annual damatge cost. As an example, for ~th a 0 - LN- F of Fig. b For. perfect information on the probability distribution of the annual flood level. Presumably the true. distribution will be lognormal or normal with equal likelihood. Given that the distribution is lognormal, the probabilities Of flooding are 0. and 0, for alt. The expected total costs for the respective ~ltematives are ·.. million ·. llb Decision tree for perfect information on distribution of annual flood level: Hence.
a 2 is optimal if the true di~tiiGLition is lognormal. Similarly, if we assume that the true distribution is normal. the probabilities of floOding are0. The expected total cos~s are t~en E C.. the maximum fund that may be spent to v. JZ excerpted from Davis and Dvoranchik, A 5CO.. foot bridge is proposed over the flood dikes at Rillito Creek, near Tucson. The bridge will rest on four piers, each supported by 25 piles. Part of the bridge may be lost in a flood as a result of scour undercUtting the piles. If this occurs, the cost is estimated to be S , Suppose p" is the probability that the bridge will need to be replaced in a year. The expected annual replacement cost is thus ,p". the objective function is the sum of the cost for the pile foundation and the expected replacement cost, given. To determine p,, the annual maximum stream flow is first studied. Based on past records of stream flows, the annual maximum flow Y may be c. utio~, the post~riorjoint distn~ution of these parameters is.
r exp f'j see Table 8. where x and s2 are the sample mean and variance of the logarithm of the n stream flow values. Hence, incorporating the effect of uncertainties in ·the distribution parameters, ~·e compute the probability of failure p, corresponding to a design pier depth has ~. I ' OUA. The results are plotted in Fig. E as a function of the design depth h. The optimal pier depth based on 10 years of data occurs at The optimal cost appears lo 'decrease with the increasing length of available data period. In the extreme case, when there are in~ite y~s· of data, no uncertainties 18 should exist in A. and C. I Years 5 10 20 40 x 3. S 12, hopt ft This approach assumes that the ·parameters of a probability distribution are random variables. The uncertainty of a parameter is modeled·by the corresponding prior or posterior distribution. If a point estimate of the parameter is desired, it can be formulated also as a problem of decision analysis.
Consider the decision tree in Fig. Depending on the particular choice of the estimator lJ and the actual value of the parameter 8, a prediction error will result, generally followed by a loss. By modeling this estimation process in the context of~ecision analysis, the Bayes point estimator may be determined such that the expected loss associated with the prediction etror is minimized. Mathematically, if lJ is the estimator of a parameter whose actual value is governed by the distribution f 8 , then the expected loss resulting from the error in prediction is L.. value Estimator 9 of 9 IPredic:tianl Error Loss Figure 1. Generally, this loss function is expressed in terms of the prediction error, namely 9- 6.
Since the Bayes estimator 6 minimizes the expecte~ loss, it s~ould satisfy the following equation: -,. E is given by. Applying Eq. in the case of a linear loss function. the Bayes estimator 9 is the median of Suppose the loss function is g O. EXAMPLE 1. except that for the small region where 01 5. e L theri: is no loss due to prediction error. the loss function as shown in Fig. As shown in the above examples, the mean, median, mode, or perhaps other values see Problem 2. Unlike the estimators in classical statistical ~estimation in which properties of unbiasedness, · sufficiency, and efficiency are used to define the quality of an estimator, Bayes point estimators are determined on the basis of minimizing the Joss associated with prediction errors.
Physically, the Bayes poin estimatOI: is more meaningful for engineering purposes, and consistent with the decision maker's objectives. iqlal sample size in statistical sampling. In Chapters 5 and 8 l'lf Vol. A larger sample size, of course, involves a higher cost; naturally, the optimal sample size will involve a trade-off between accuracy and the cost of sampling. Suppose X is a random variable with a paramett:r 8 to be estimated from sampling observations. The decision model for determining. the optimal sample size is shown in Fig. Two phases of decision may be required. First; at B, a decision on the sample size n is required. is observed; the appropriate estimator for 8 is selected at C. JiXJ {x} d{x} J,,. This function represents a quadratic loss because of an error in the estimation of Jl and a linear sampling cost.
u arc the posterior mean and variance of J. dE Lin. which gives Bop! n value of JL However, the optimal estimator of Jl is not always given by the posterior mean. For example, n. if the less f! mction assumed. in Eq. l, will not. be equal to p. u , depends on the distribution of the underlying random variable population. For the case in which the. population is Gaussian with knov. Therefore, in this case, the expected loss depends only on the sample size n, but not on any sample statistics of the '. For a Gaussian population X with known standard deviation u, if we substitute Eq.
Resendiz and Herrera studied a settlement problem for a foundation in which they determined the optimal design pressure see Problem suppose the loss due to error in a distance prediction is proportional to the square of the error in inches. The standard error in each measurement with the proposed device is I inch. Determine the optimal number of measurements for the following cases: '. a No orior information is known on the distance. b A prlor information indicated that the distance is N IOOO. estimation of p. With Eq. as N IOOO, 0. c If the prior information on. is N , 0. Hence, the best strategy in this case is not to make any additional measurements, but to estimate the distance using the available prior information giving an estimated distance of inches. If DO in the stream is normally distributed with mean ~ and : standard- deviation 0.
where {J is the estimator of the mean DO concentration; lOO u.. Substituting the given loss function in Eq. E Lin, {x}. pl is the posterior mean of"' The corresponding minimum expected loss is E[Lin, {x}, fJ. Equation 2. The occurrence of cracks along a highway pavement may be modeled as a Poisson process with a parameter m, which is the mean occurrence rate of cracks. From a previous inspection of similar pavements, the parameter m is estimated to have a mean of 2 per mile with a c. Additional inspection may be conducted, at a cost ofS20 per mile, to improve the accuracy of the estimated value of m. Assume that the prior distribution of m is given by the conjugate distributio! l to the Poison random variable. Determine the optimal distance t in miles to be inspected that will minimize the expected total cost function. ll, we sec that the conju. gate concept requires a gamma distribution for the parameter m.
Hence, the prior distribution of M will. bc of the form ' The parameters v' and It can be evaluated from ·~- -. The decision model for this example is shown in Fig. The corresponding optimal loss at node C becomes E[Lit, x; 9. ar" M.. Hence, from Eq. Sc;ning the derivative to zero, ~e-obta. Ranking the feasible alternatives in a decision tree requires a scale for quantifying the degree of preference among the attributes. A well estabii'shed scale of measure may exist for some of the attributes such as monetary value; however, there is no obvious · way of quantifying the values for the majority of these attributes.
The problem of value measurement is further complicated when the consequences in the decision tree require the evaluation of a combination of attributes. In order to establish a uniform scale for measuring the overall value of an alternative, the concept of utility may be introduced. Utility is defined as a true mea,sure of value to the decision maker. Utility theory provides a framework whereby values may be measured, combined, and compared consistently with respect to a decision maker. Therefore, if the utility values of all the alternatives are available, the alternative with the highest utility value will be preferred. een ~-and ·B" · A ~B denotes. is preferred at least as much as B" ·.. A- B, A~B. This indifference may be presented schematically with the following lottery: ~~B ~: It is obvious from the lottery that if p is 0, getting C is certain, hence, B is ol:learly preferred. Whereas if p is 1, the lottery implies getting A for sure, and -. B is not pr~ferred. will emerge.
In mathematical tcrminologi, B is the cercainty equivalent of the latter lottery. A lottery and its certainty equivalent are interchangeal:lle without affecting preference. This is obvious from iii that shows the two are equally preferred. v M onotonicity. A series of branches may be replaced by a single branch as, for example, · ~A ~. Most of these axioms are obvious, yet they form the backbone "of the field of utility theory. A utility flinction quantifies the· order of preferences. Mathematically, the function represents a mapping of the degree of preference into the real line, thus permitting preference to be expressed numerically.
For example, the utility of the following lottery between A and C is ·. a It can be demonstrated. that utility function is consistent with the axioms of utility theory. As an example, consider the aiiom of monotonicity. If we apply the property of the· utility function in Eq. In other words, if a consistent set of utility values u· are transformed to u' through the following.. decision maker's original preference. This property can be easily proven as follows. Suppose that £ 1 , £ 2 , £ 3 are in increasing. order of preference with respective utility values u E 1 , u E 2 , and u E 3. From the continuity axiom of utility theory, a value of p exists such that Multiply each term in Eq. Since the values of a and b in the above transformation can be arbitrarily assigned, it follows that in determining a consistent set of utility values, the first two values can be arbitrarily assigned, as illustrated fu the subsequent section.
As long as all other utility values are consistently determined with respect to. these two reference points, all the utility values will form a consistent set, In other words, utility values are not unique. According to Eq. Suppose there are three events, say E 1, Ei, E~. to which utilities values are to be ;:. First, arrange the events in decreasing. This is similar to the assignment of 1. To' determine the utility. of E 2 ', relative to those of E 1 and £ 3 , the decision maker is offered a choice between the following lotteries:.
Staelvon Holstein, :The probability wheel is a disk with two sectors, one blue and the other red, with a fixed pointer in the center of the disk. relative sizes of the two sectors, which in turn change the probabilities of the pointer landing on either sector as the disk stops spinning. In determining the value of p. is spun. If ~he pointer ends up in. up in.. The red sector in the wheel is then increased until indifference is observed. At that. point, the fraction of the red sector in the wheel is simply the probability pat which there is indifference between L 1 and L 1.
This value of pis used only to determine u £ 2 ; it is independent of the probabilities in the decision tree. Iq a complex decision tree, a large number of events will require the assignment. of utility values. The following procedure may be used to systematically determine the relative utility values. of n ~yents,. are taken as the reference points for the lottery. the value of p obtained each time will very likely be different. has been determined. However, to provide cross checking on these values, repeat Step 3 using u E 1 and u E. If any inconsistencies are found in the above procedure, repeat the process until all utility values agree satisfactorily. If time permits, steps 5 and 6 may be repeated with E. As an alternative,· multi- · attribute decision analysis sec Section 2. Iow S0 2 ; iii low CO, high S0 2 , iv high CO, high SOl concentrations.
It is obvious that the desirability of the events would vary inversely with the concentration of pollutants. Furthermore, the decision maker feels it is more urgent to reduce sol than co at this time. and start with the questioning process. Suppose the decision maker is indifferent between each of the following pairs of lottery, namely: · ·· Then, from Eq. to provide cross checks on these utility assignments: the decision maker is asked for the value ofp that he"would"be indifferent between. Therefore, inconsistency exists and the decision maker is asked to reevaluate his assignment of probabilities. In the second round. after careful consideration. the decision maker is found to be indifferent between each of the following three pairs of lotteries: ~EI. However, monetary value is not always a consistent measure of utility; that is. the preference order of the decision maker may depend on the magnitude or amount of money actually involved.
Consider the following. A decision maker is asked to choose between alternatives I and II from the following pair oflotteries·:. In lottery 1, the outcome will be either E 1 or E 2 with equal likelihoods; whereas in lottery II. Observelhat the expected monetary-values are the same for the two alternatives in all cases. The above illustrates the. In general, therefore, ll utility function involving monetary. values for a specific decision maker must be esublished. The procedure is illustrated by the following example. His utility function of money over this range is to be determined. To establish the utility scale, the utilities ofSIOO and SO are assigned arbitrarily to be 1.
Since the utility value of 0 is not assumed for the lowest dollar value involved, the utility function will take on negative values. The first pair of lotteries is presented to the decision maker as follo~s: With lottery I, the engineer stands a SO chance of gaining SlOO or nothmg; whereas he will get Sx 1 for sure with lottery ll'At what value of xj will the f'vo lotteries be indifferent? the decision maker is given the following pair of lotteries; · A value of x 4 is required to ~tablish indifference. u O ;;,·u. Figure E graphically shows the utility function fitted through the six utility values.
In Example 2. lr values so that indifference is achieved. In earlier sections, the events £ 1 were fixed and the decision maker was asked to determine the probabili~ p s0 that indifference is established. Clearly, therefore, there is more than one method of constructing the required utility function. where IX and fJ are constants. erting Eq. where IX' and fJ' are also constants. Since u d is a utility function, applying the property of linear transformation, the dollar v~ue d is also. a consistent utility function. In other words, monetary value may be a valid utility furiction for a large finn. Figur~ E2Jl A utility function of money. Mathematically, the construction of a -utility function involves the trans- formation of the values of these continuous variables'mto respective consistent scales in the utility space, representing the decision maker's actual. preference for values of the original variables. As an illustration, consider the decision analySis of the development of the aii-port facilities in Mexico City de Neufville and Keeney, and M.
S at the University of Illinois. at the Mapua Institute of Technology. 검색 이미지 지도 Play YouTube 뉴스 Gmail 드라이브 더보기 캘린더 번역 도서 쇼핑 Blogger 금융 사진 문서. Account Options 로그인. 내 라이브러리 도움말 고급 도서검색. 인쇄판 구매하기. 교보문고 알라딘 인터파크 아마존 영문 예스24 리브로 반디앤루니스 도서관에서 찾기 모든 판매자 ». Google Play에서 도서 쇼핑 세계 최대 eBook 상점을 둘러보고 웹, 태블릿, 휴대전화, eReader에서 독서를 시작해 보세요. 지금 Google Play로 이동 ». Probability Concepts in Engineering: Emphasis on Applications to Civil and Environmental Engineering, 2e Instructor Site. Alfredo H-S. Ang , Wilson H. Apply the principles of probability and statistics to realistic engineering problems The easiest and most effective way to learn the principles of probabilistic modeling and statistical inference is to apply those principles to a variety of applications. Key Features in this 2nd Edition: A new chapter Chapter 5 covers Computer-Based Numerical and Simulation Methods in Probability, to extend and expand the analytical methods to more complex engineering problems.
New and expanded coverage includes distribution of extreme values Chapter 3 , the Anderson-Darling method for goodness-of-fit test Chapter 6 , hypothesis testing Chapter 6 , the determination of confidence intervals in linear regression Chapter 8 , and Bayesian regression and correlation analyses Chapter 9. Many new exercise problems in each chapter help you develop a working knowledge of concepts and methods. Provides a wide variety of examples, including many new to this edition, to help you learn and understand specific concepts.
Probability Concepts in Engineering V1 2 Ed - Ang y Tang PDF. Uploaded by Julian Trejos. Document Information click to expand document information Original Title Probability concepts in engineering V1 2 ed - Ang y Tang. Copyright © © All Rights Reserved. Available Formats PDF or read online from Scribd. Share this document Share or Embed Document Sharing Options Share on Facebook, opens a new window Facebook. Did you find this document useful? Is this content inappropriate? Report this Document. Copyright: © All Rights Reserved. Available Formats Download as PDF or read online from Scribd. Flag for inappropriate content. Download now. Save Save Probability concepts in engineering V1 2 ed - Ang For Later. Original Title: Probability concepts in engineering V1 2 ed - Ang y Tang. Jump to Page. Search inside document.
You might also like Ang and Tang Solutions. Jack Benjamin, C. Cornell-Probability, Statistics, And Decisions for Civil Engineers -McGraw-Hill Companies Ang y Tang ProbabilityConceotinEngineering. Probability concepts in engineering planning and design Vol II. Proyecto Final Proyecto Final Grupo 1. Proyecto 3 - Grupo 1 V2. Ejercicio 1. Proyecto 4 - Grupo 1. Ejercicio 2. Proyecto 5 - Grupo 1. Seminario 6. Charla Técnica Nº 3. Charla Técnica Nº 2. Charla Técnica Nº 6. Charla Técnica Nº 1. charla 3. Charla 4. Seminario 2. Seminario 3. Seminario 5. Seminario 4.
Probability Concepts in Engineering V1 2 Ed - Ang y Tang PDF,Account Options
WebProbability Concepts in Engineering Emphasis on Applications in Civil & Environmental Engineering 2NDEDITION Prob ab iIity Concepts in Engineering* Emphasis on WebProbability Concepts in Engineering: Emphasis on Applications to Civil and Environmental Engineering [2 ed.] , X. Apply the principles WebOct 25, · Probability concepts in engineering V1 2 ed - Ang y blogger.com - Free ebook download as PDF File .pdf) or read book online for free. Scribd is the world's largest WebProbability concepts in engineering planning and design by Ang, Alfredo Hua-Sing, Publication date Openlibrary_edition OLM Openlibrary_work WebSep 24, · The second edition of this well-known book (previously titled Probability Concepts in Engineering Planning and Design) by Alfredo Ang and Wilson Tang, two ... read more
and start with the questioning process. In these latter cases, the selection of the.. p -' ln p-. A third alternative is to perform seeding experiments before deciding on whether or not to seed. Accordingly, the max EMV criterion may not always be the proper criterion for selecting alternatiyes that will reflect the decision maker's actual preference.
In reality each ofthe above assumptions may not be totaly valid For example, depending on the materia ofthe beam and the magnitude ofthe load the behavior may not be linearly elastic. ll, we sec that the conju. Probability Papers 7. Design B is based on modem concepts and will reduce the cost of the building to Sl million. Account Options 로그인. Deteriorating Systems Problems for Chapter 3.
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