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Meier-Zabell

Course: BEN-ISRAEL 711, Fall 2008
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Peirce Benjamin and the Howland Will Paul Meier; Sandy Zabell Journal of the American Statistical Association, Vol. 75, No. 371. (Sep., 1980), pp. 497-506. Stable URL: http://links.jstor.org/sici?sici=0162-1459%28198009%2975%3A371%3C497%3ABPATHW%3E2.0.CO%3B2-G Journal of the American Statistical Association is currently published by American Statistical Association. Your use of the JSTOR archive indicates your...

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Peirce Benjamin and the Howland Will Paul Meier; Sandy Zabell Journal of the American Statistical Association, Vol. 75, No. 371. (Sep., 1980), pp. 497-506. Stable URL: http://links.jstor.org/sici?sici=0162-1459%28198009%2975%3A371%3C497%3ABPATHW%3E2.0.CO%3B2-G Journal of the American Statistical Association is currently published by American Statistical Association. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/astata.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers, and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take advantage of advances in technology. For more information regarding JSTOR, please contact support@jstor.org. http://www.jstor.org Tue Apr 1 14:55:42 2008 Beniamin Peirce and the Howland PAUL MElER and SANDY ZABELL* Will The Howland will case is possibly the earliest instance in American law of the use of probabilistic and statistical evidence. Identifying 30 downstrokes in the signature of Sylvia Ann Howland, Benjamin Peirce attempted to show that a contested signature on a will had been traced from another and genuine signature. He argued that their agreement in all 30 downstrokes was improbable in the extreme under a binomial model. Peirce supported his model by providing a graphical test of goodness of fit. We give a critique of Peirce's model and discuss the use and abuse of the "product rule" for multiplying probabilities of independent events. that the earlier will should be recognized. But this the Executor, Thomas Mandell, declined to do. Not only did he claim that the later will governed, but he also contended t h t txvs-of the three signatures on the earlier will were tracings from the third, thus providing an independent ground for invalidating that mill (see Appendix I ) . Hetty Robinson retained distinguished counsel and KEY WORDS: Charles S. Peirce; 19th-century mathematical sued.' Thomas blandel retained equally distinguished statistics; Law and statistics; Handwriting analysis; Independence; counsel and defended. Between them, both sides called Product rule. as witnesses some of the most prominent academicians of the day. Oliver Wendell Holmes, Sr., Parkman Pro1. INTRODUCTION fessor of Anatomy and Physiology in the Medical School On July 2, 1865, Sylvia Ann Howland of Xew Bedford, of Harvard University, examined the contested signaMassachusetts, died, leaving an estate of $2,025,000, tures under a microscope, as did Louis Agassiz, and and a single heir-her niece Hetty H. Robinson, who testified (for Robinson) that he could find no evidence had lived with her for some years. Two million dollars is of pencil marks such as might have appeared in tracing. a respectable sum, even by today's inflated standards, (The cross-examination of Professor Holmes mas both and in 1865 it was certainly a prize worth contesting. brief and painful; see Appendix 2.) Among those testifyThe Howland will, dated September 1, 1863, and a ing for the Executor were Benjamin Peirce, Professor of codicil of November 18, 1864, left half the estate to a Mathematics a t Harvard, and his son, Charles Sanders number of individuals and institutions. I t provided Peirce, who was later to do important work in several that the residue was to be held in trust for the benefit of branches of philosophy. The Peirces mere among the Hetty Robinson and, a t the time of Hetty's death, to be first to contribute to the development of mathematical distributed t o the lineal descendents of Hetty's paternal statistics in the United States; a recent survey of their grandfather, Gideon Howland, Sr. Hetty's father had work in this area is given in Stigler (1978, pp. 244-251). died a month earlier, leaving her $910,000 outright and Professor Peirce undertook to demonstrate by stathe income from a trust of $5,000,000. Although Hetty tistical means that the disputed signatures on the second Robinson was thus already a wealthy woman, she page of the will were indeed forgeries. His method was claimed a right to inherit outright the entire estate of to contrast the similarities between one of the disputed her aunt, providing as evidence for this claim a n earlier signatures (referred to in the trial record as signature mill, dated January 11, 1862, mhich left the entire estate # 10) and the unquestioned original (referred to as t o Hetty with instructions that no later will was to be signature # 1) with the lesser degree of similarity to be honored. found in pairing 42 other signatures penned by Sylvia Family feuds are relevant here, as is a claimed agree- Ann Howland in her later years on other documents. ment between Hetty and her aunt to leave mutual wills (See Figures A and B.) I t is this analysis with mhich excluding Hetty's father. For our purposes, it is enough we shall be concerned. to observe that Hetty Robinson had a n arguable claim The Peirce analysis, as we shall see, is a t once ingenious and in some respects naive. I t leads to a distributional * Paul Meier is Professor of Statistics, The University of Chicago, problem, discussed in Appendix 4, which is nontrivial Chicago, IL 60637, and Sandy Zabell is Associate Professor of and has some interest in its own right. The data available Mathematics, Northwestern University, Evanston, I L 60201. 1865. Testimony was taken in 1867, decision rendered october 1868 ( ~ ~ v. b ~ i ~ ~20~ ~ l ~~cas.1027). ~ d l ,d ~, A copy of the trial transcript is available in the New Bedford Public ~ 2 and 3 is ~ ~ i and the testimony ~ the ~ b ~ in text ~ ~ extracted from that source, @ Support for this research was provided in part by NSF Grant SOC 76-80389. The authors are grateful to a number of friends and colleagues with whom they discussed aspects of this caseP. ~ i a c o n i s ,S. Fienberg, J. Langbein, P. McCullagh, J. T u k e ~ * and H. Zeisel, among others. Special thanks are due to David Wallace for extensive discussion concerning the inference problem that arose, and Stephen Stigler for helpful correspondence on the historical background. The staff of the New Bedford Public Library were especially accommodating in locating and arranging for the copying of unique source documents and exhibits, and to them the authors express their special gratitude. , Journal of the American Statistical Association September 1980, Volume 75, Number 371 Applications Section 497 498 Journal of the American Statistical Association, September 1980 A. The Unquestioned Original Signature (no. 1) and the Two Disputed Signatures (nos. 10 and 15) human affairs. We should not be surprised, therefore, to find such imputations as a matter of course in tmhe present problem. 2. THE PROBLEM Let us start with the evidence as given, making assumptions as needed, and return later to judge which seem plausible and which not. The evidence consisted of photographic copies of 42 signatures, in addition to the original (# 1) and one of the purported copies (# 10). Suppose we are given some index of agreement between signatures. I n the present case Professor Peirce chose the number of coincidences-in length and position-among the 30 downstrokes in each of a pair of signatures, when the two were superimposed. We might then let XI, . . . ,XqZ be the values of that index obtained in the record is sufficient to allow some interesting in matching each of the 42 uncontested and uninvolved analyses, but tantalizing for its failure to include details signatures with # 1, and let Y be the value of that index of major relevance for the problem a t issue. However, in matching # 10 with # 1. We could then pose the probas we proceed, it is well to keep in mind that no organized lem as one of judging whether Y could reasonably have theory of statistical inference existed a t the time, and been drawn a t random from the same population that that, for example, casual imputations of independence gave rise to the sample of size 42. Since, in this case, Y = 30, that is, # 1 and # 10 coinwere made almost automatically by those seeking to bring cide in all 30 downstrokes, and the X, are presumably considerations of probability to bear on problems of B. Some of the 42 Comparison Signatures Meier and Zabell: Benjamin Peirce and the Howland Will 499 1 . C. S. Peirce's Table of Coincidences Coincidences Number of Pairs all smaller than 30, we might assign a significance probability of 1/43 = 2.3% to this outcome. This is a small enough probability to raise suspicion, but far from over~ h e l r n i n g (If we had available the individual X , values .~ (which we do not) we might be willing to proceed a little more parametrically. We can determine (see Table 1) that the average index in random pairings of the 42 signatures is near 6 with a standard deviation of about 2.5. Thus Y is nearly 10 sample standard deviations away from L and, if we believe that the population f sampled is a t least approximately normal, the discrepancy would be judged significant a t an extreme level.) Professor Peirce proceeded in a different way and came to the conclusion that the odds against such a coincidence as that between signatures # 1 and # 10 were astronomical (of order loz0).T O see how he arrived a t this striking result, let us follow the order of testimony in the case. 2.1 Testimony of Charles Sanders Peirce - Number of Strokes The son, Charles Sanders Peirce-then a member of the staff of the United States Coast Survey-testified on the work he carried out under the direction of his father. Charles was provided with photographic copies of signatures # 1 and # 10 (the third signature is not mentioned in the testimony) and with copies of 42 signatures of Sylvia Ann Howland on leases and other documents. Following instructions given to him by his father, Charles undertook to match every possible pair of the 42 uncontested signatures and to observe the agreement between each pair in respect to the number out of 30 identified "downstrokes" in which they were in agreement. There being ( y ) = 861 possible pairs, and 30 downstrokes to be checked for each, this meant a total of 25,830 comparisons, for which Charles reported agreement in 5,325 or 20.6 percent of them-very nearly 1 in 5. More particularly, he gave the distribution of number of agreements as shown in Table 1. Thus, for example, 15 pairs each had two coincidences out of 30, for a total of 2 X 15 = 30 strokes coinciding. The precise details of the matching process and of the criteria for judgment of agreement were only briefly touched on in C.S. Peirce's testimony: Q. What was the standard . . . of coincidence? A . I considered that two lines coincided whenever they coincided as well as . . . nos. 1 and 10, in those same lines. that the line of writing should not be materially changed. By materially changed, I mean so much changed that there could be no question that there was a difference in the general direction of the two signatures. 2.2 Testimony of Benjamin Peirce C.S. Peirce was followed to the stand by his father, who testified as to his own examination and comparison of # 1 and # 10. The coincidence is extraordinary and of such a kind as irresistably to suggest design, and especially the tracing of 10 over 1. There are small differences . . . but the differences are such as to strengthen the argument for design. . . . The coincidences are manifestly those of position; while the differences are those of form. Coincidence of position is easily effected by design, and can be tolerably well accomplished by an unskilful hand. I t especially belongs to the downward stroke. . . . I t involves a close agreement in the average position of the stroke, an agreement in its slant, and no extravagant diversity of curvature. But coincidence of form is exceedingly difficult. . . . In a long signature, complete coincidence of position is . . . an infallible evidence of design. The mathematical discussion of this subject has never, to my knowledge, been proposed, but it is not difficult; and a numerical expression applicable to this problem, the correctness of which would be instantly recognized by all the mathematicians of the world, can be readily obtained. Later, under cross-examination, Charles added some further details : I made it a condition that in shifting one photograph over another in order to make as many lines as possible coincide,%Onemight try to improve on this a little by considering the probability that out of 44 signatures, a given pair will have the greatest index of coincidences, that is, (424)-1 = ,0011. (Strictly speaking, we are not given the number of coincidences between either signature # 1 or #10 on the one hand, and any of the 42 other signatures on the other, but presumably none of these even approaches the 30 of # I and # 10.) The appropriateness of such a significance test is an interesting issue, but one that we will not touch on. Professor Peirce went on to describe his son's work and, without qualification, asserted that since the overall proportion of matches was very nearly one-fifth, the probability of finding 30 matches in a given pair of signatures was one in 530or "once in 2,666 millions of millions of millions." (Kote: There is some error here, since 530= 9.31 X loz0.)Lest the point be missed, Peirce added : This number far transcends human experience. So vast an improbability is practically an impossibility. Such evanescent shadows of probability cannot belong to actual life. They are unimaginably less than those least things which the law cares not for. When asked the purpose of his son's display of the distribution of the number of coincidences, Peirce ex- 500 2. Benjamin Peirce's Table (and chi-squared computations) Journal of the American Statistical Association, September 1980 no cases of 0 or 1 agreement, with 9 expected-and the considerable excess of outstandingly good matches-20 (with 13 or more agreements), with less t'han 3 expected. Number of I we calculate the usual goodness-of-fit chi-squared f Agreements Obs. Exp. (0 - E ) (0- E)*/E for this frequency table we get X2= 170.7 on 12 df. We can improve this a little by choosing p to minimize 0 9.1 -9.1 9.1 chi-squared, but the minimizing p = .211 actually - 14.0 6.7 2 15 29.0 worsens things in the middle range of the graph and 67.6 29.4 12.8 3 97 gives a X2= 135.3, which remains a n exceedingly poor 4 131 114.1 16.9 2.5 fit. (The appropriateness of the chi-squared statistic is discussed in Sec. 3.2.) If one compares Peirce's graphical test of fit with the chi-squared statistic, it is obvious that the visual impact of the graphical display is to weight all deviations from the expected values equally, while chi-squared weights such deviations relative to the magnitude of each expected value. For this reason, discrepancies between the observed and expected values in the tails, even if they had been plotted, might not have appeared to be more serious than those in the middle of the distribution, while it is precisely the contribution of the tails to chi-squared that makes the plained that latter significant. Clearly the most frustrating aspect of this data set is I t was ordered by me in order to obtain as severe a test as possible from observation for the criticism of my mathe- its lack of detail with regard to the 20 good matches. matical deductions. Were even one of them to show 30 agreements, or even 29, the persuasiveness of the mathematical evidence To facilitate comparison, Peirce provided a table, recapitulating his son's counts, together with the ex- would evaporate. Indeed, as shown in the earlier table, we can calculate from the data given that the total pected number of counts under the binomial model number of agreements in these 20 cases was 288, or an average of 14.4 for those signatures with 13 or more agreements. Thus, although one or two very high values remains a numerical possibility, it seems likely that few (See Table 2. The table given by Peirce omits the 15 if any values were as high as 20. cases with two or fewer coincidences, and the 20 with Unfortunately for the record, Professor Peirce,s more than Peirce commented : demeanor and reputation as a mathematician must have This is a species of comparison familiar to mathematicians been sufficiently intimidating to deter any serious and such a coincidence as above would be regarded as quite extraordinary, and a sufficient demonstration that the true mathematical rebuttal. He was made to confess a lack of any general expertise in judging handwriting, but he was law of the case was embodied in the formula. 71 A graphical display of the comparison was also provided, which is reconstructed in Figure C according to the description given in the trial record (the actual display was omitted from the trial record). 3. CRITIQUE OF PEIRCE'S MODEL C. Distribution of Coincidences FREQUE401 200 --- OBSERVED - EXPECTED FREQUENCY FREQUENCY 150- 3.1 Goodness of Fit How convincing is Peirce's model and his evidence for it? No formal theory of goodness-of-fit testing ex- leeisted a t the time, and Peirce can scarcely be faulted for failing to provide one. Still, the quality of fit can reasonably be questioned. Peirce's graph shows only the middle range, and even here there appears to be a con- 50sistent trend in the deviations. What the graph does not show is the substantial absence of really poor matches- - 'The table in the trial record gives 41 cases as "undistributed," but it is clear from Peirce's testimony immediately following that 35 was meant. ("My son had also 35 undistributed cases . . . .") "1 0 I I I I s I I I 10 " i I I I I 1 I I 20 1 1 1 I 25 8 1 I 30 15 NUMBER OF AGREEMENTS Meier and Zabell: Benjamin Peirce and the Howland Will 501 dependence, on the other hand, remains in full force.) That test, however, as noted before, yields only a strongly suggestive significance probability, but nothing a t all as compelling as Peirce's calculation. For all that one can raise questions about Peirce's model and analysis, it remains to consider the extent to which the evidence available really contradicts it. Although the conventional chi-squared test clearly rejects the binomial model, it is not a t all clear that the chisquared statistic should have even approximately the familiar distribution. Under Peirce's model the probability that a random pair of true signatures should match in k stokes is indeed binomial, as claimed, and thus the expected fraction of pairs with k agreements is also binomial. However, the (p)= 861 pairs are generated from only 42 signatures and are not a t all the same as 861 independent trials from a binomial population. I t is to be expected, therefore, that the relative frequencies will show variability greater than multinomial and that the chi-squared statistic will also exhibit greater variability. T o assess more rigorously the possible lack of fit of Peirce's model we turn to consideration of the rather large excess of pairs with over 12 matches (20 observed, only 2.7 expected). The hlarkov inequality (e.g., see Billingsley 1979, p. 65), of which the familiar Chebyshev incquality is a direct corollary, states that for any nonnegative random variable Z, one has not cross-examined a t all on the numerical and mathematical parts of his t'estimony, let alone made to list the 20 high values (see Appendix 3). 3.2 Model Assumptions The frequent paralysis of the law before a demonstration of mathematics has long been noted. An extreme view, expressed by Laurence Tribe in his 1971 Harvard Law Review article "Trial by Mathematics" (Tribe 1971) is that laymen are too readily impressed and misled by such presentations and that they should therefore be excluded-at least in jury trials. Whatever one thinks of this position as a generality, Peirce's model, as well as his data analysis, deserves closer examination than it received during the trial. One might, for example, question the assumptionrequired for the binomial model-that the probability of a match should be the same a t each of the 30 positions. One can easily imagine that certain strokes-perhaps those early in the signature-would have a higher probability of matching than do others. The assumption of equal probability could readily have been checked in the course of C.S. Peirce's work, but it seems never to have been questioned. (Of course, the effect of such inequality would be to reduce the variance in the number of matches, and thus to make the probability of 30 matches even smaller than that quoted.) Far more serious is the implicit assumption of independence. One would expect that agreement in, say, positions 1 and 3, would make far more likely a n agreement in position 2. The effect of such dependence would be to increase the variance and thus increase the probability of 30 matches over that quoted, quite possibly by orders of magnitude. T o the extent that the data bears on this point, it does indeed suggest the possibility of dependence by exhibiting what appear to be far too many cases with over 12 matches to agree with the binomial model. A final critique of Peirce's discussion is the absence of any comment on correlation between signatures made close in time. I t is a t least plausible that signatures late in life would differ substantially from those made a t a younger age, and also plausible that signatures made a t a single sitting would be more similar than those made days or weeks apart. (Similar problems arise in the statistical analysis of authorship, where both style and vocabulary may depend on the writer's subject matter and may vary over his lifetime; see Mosteller and Wallace 1964, pp. 18-22, 195-199, 265-266.) Since many of the 42 signatures used in the study were on leases and other dated documents, this question could also have been examined, but there is no evidence that it was considered. I n part these challenges to the fairness of Peirce's test can be met by the nonparametric procedure suggested earlier. The questions of uniform probability of matching in each of the 30 positions, and independence between them, would then be (The irrelevant,. problem of time Taking Z = the number of pairs out of the 861 showing 13 or more agreements, and t = 20, we find P r ( Z 2 t ) 5 2.7/20 = .14-not in itself sufficient reason to reject the Peirce model. A more refined analysis, however, gives fairly convincing evidence that the Pcirce model does not fit the data (see Appendix 4). 3.3 Independence and the Product Rule Peirce's unargued and possibly unwarranted assumption of independence should occasion no surprise. I t was often takcn for granted by thoughtful and astute scientists of that era and later oncs-not entirely excepting the scientific community of today. For example, Sinlon Ncwcomb wrote in The Universal Cyclopedia (1900, s.v. "Probability, Theory of") : The mathematical rule for determining probability in such a case [when the concurrence of a large number of circum, stances is necessary to the production of an event] is that the probability of the concurrence of all the events is equal to the continued product of the probabilities of all the separate events. As one example, suppose that a law requiring the concurrence of the two houses of Congress and the President were as likely as not to be rejected by any one of them, and that each one of the three authorities formed his own opinion independently of the other two. Then the probability of its passing all three would be 3 X 3 X t = t . The product rule is misstated here, independence being only introduced subsequently in the context of a n example. And while Newcomb has the mathematics right in his example, the application, although it can be given 502 Journal of the American Statistical Association, September 1980 the correct interpretation, is exceedingly apt to be given an incorrect one. For let us agree that each branch of government arrives a t its own opinion independently of the other two on each law, where "independently" is given the common meaning of deciding in ignorance of the deliberations or decision of the others. I t does not a t all follow from that that the decisions will be statistically independent. Thus, if a law on capital punishment is under consideration, all might be influenced by public opinion, a variety of lobbies, or the recent report of some heinous crime. The failure of each to directly influence one another does not protect them from being simultaneously influenced by external events. The confusion of this ordinary meaning of "independence" with "statistical independence" remains a common error today and, although Newcomb himself may have been clear about the matter, his example here seems especially unfortunate. For while Newcomb may just have been careless, skimming over a point obvious to himself, the statisticallayman who read him could easily go astray. This was certainly the case for a noted expert on forgery, Albert, S. Osborn (1929, pp. 225-234), whose text Questioned Documents cites "Professor Newcomb's rule" with approbation, but whose illustrations of it show no awareness of the need for independence. In one instance, Osborn supposed an inches tall, with blue individual sought who was 5 feet 11% eyes and brown hair, had lost his left thumb and the lower part of his right ear, and had a distinctive mole, tattoo, and scar. Osborn assumed for the sake of argument that the first of these traits had a probability of &,the second 5, the third 4, and all the rest 1/200, and concluded that the probability of their joint occurrence was 1 in 38,400,000,000,000. There is, of course, no reason to suppose a priori that height, eye color, and hair color are independent traits, although in this instance possible mutual dependence might not be too serious. Surely, however, the occurrences of a tattoo, a scar, and a missing thumb and ear would be highly dependent. (A more recent and notorious instance is the celebrated Collins case.4) This intuitive but incorrect interpretation is not entirely off the mark, however, in a case-such as the present one-in which the probability of the evidence, given the alternative (i.e., close agreement in the signatures, given a deliberate forgery), is taken to be very near one. For in that case the likelihood ratio P r (evidence I null hypothesis) Pr (evidence 1 alternative) - significance probability 1 and the posterior odds for the null hypothesis thus reduce to (significance probability) X prior odds . Hence, in this case, if the prior odds are not extreme, the posterior odds will indeed be of the same order of magnitude as the significance probability. Another problem affecting the apparent strength of the evidence, however, comes from a very different direction. In framing the problem Peirce quite reasonably divided the possibilities into two categories-either Sylvia Ann Howland wrote both signatures, or she didn't. I n the latter case we are left to presume that Hetty Robinson had a hand in the matter. However, there are in fact many alternatives, which, although not very plausible, still have prior probabilities well in excess of 5-30. For example, Sylvia Ann Howland might have been in the habit of tracing her own signature. Or perhaps the Executor, devoted to the destruction of Hetty's claim (and her reputation), might have stolen the original "second page" and replaced it with a forgery. Or we could imagine that C.S. Peirce, in his enthusiasm for the mathematical method, might have been overgenerous in counting agreements when comparing signatures # 1 and # l o , and more particular when comparing other signatures. Indeed, almost any outrageous possibility that one could conceive of, provided only that it be physically possible, would have a prior probability many orders of magnitude greater than 5-30. Without going into the calculations, it should be clear that the odds against Hetty's guilt cannot be assessed as anything less than the prior odds in favor of these LLoutrageous" possibilities. (The authors' intuitive assessment puts the odds in favor of some "outrageous" explanation in the or present case as a t least lom4 10-5, and thus-even if we accept Peirce's model in its entirety-the strength of the evidence against Hetty is bounded in this way.) The problem of "outrageous" alternatives and their effect on limiting extreme posterior probabilities is discussed by Mosteller and Wallace (1964, pp. 90-91). Despite the criticism made here of Peirce's model and data analysis, it stands as an example, in many ways excellent, of the early use of formal statistical methods in a scientific and social setting. Not only did Peirce formulate the problem in a plausible probabilistic framework, and calculate a proper significance probability for that model, but he provided a test of the model itself. That the test was solely graphical is in no way to his 3.4 '"Outrageous" Events There are some general criticisms that can be leveled a t the use of a n extremely small significance probability to exclude one hypothesis in favor of another. Significance probabilities are frequently misconst'rued to be statements of posterior probability when only two alternatives are under consideration (here, "Sylvia Ann Howland wrote both signatures independently" vs. "Hetty Robinson forged the second signature") ; that is, a significance probability of e against the null hypothesis comes to be regarded (erroneously) as effectively a posterior probability of 1 - e in favor of the alternative. 68 Cal. 2d 319, 438 P.2D 33, 66 Cal. Rptr. 497 (1968). Fairley and Mosteller (1977, pp. 355-379) give the decision of the California Supreme Court and a critical evaluation. Meier and Zabell: Benjamin Peirce and the Howland W i l l discredit, since no more incisive testing procedures had been developed, nor would be for many years to come. Considered against the background of his era, Peirce's analysis must be judged unusually clear and complete, even though-from a modern viewpoint,-we may regard his conclusion about the significance of his findings less compelling than he did. 4. AFTERMATH Peirce was only one among many witnesses for both sides, but from our point of view the next most interesting witness was J.C. Crossman, a n engraver. Crossman testified for the plaintiff, essentially in rebuttal to Peirce. Peirce had said that such agreement as found between signature # 1and # 10 could not occur except by design, no matter how regular the habits of the writer. Crossman, however, gave testimony about the regularity of the signatures of several individuals, most notably those of the sixth President of the United States, John Quincy Adams (who signed his name "J.Q. Adams"). I n each of these cases, Crossman testified, pairs of signatures had been found in which the matching was as good as or better than the matching of # 1and # 10. Unfortunately, the testimony does not suggest the corresponding value of p (.2 for Sylvia Ann Howland), which powerfully influences the final probabilityespecially since the number of strokes to be matched in this case is considerably less than 30. How convincing was the Peirce testimony a t the time? Or what of Crossman's, supplementing the microscopic examination of Professors Holmes and Agassiz? The court did not say directly, but its decision can be viewed as suggesting that the court was a t least left in doubt about the validity of the contested signatures. I n brief, the legal situation was as follows. As a general matter, a will may be revoked by the testator a t any time until his death, although he can by contract bind himself not to exercise his power of revocation. Thus one must look for t,he grounds on which Hett,y Robinson asked that t,he earlier will and not the later one prevail. She might have sought t,o discredit t'he lat,er will on the ground of undue influence. (Indeed, in a n earlier proceeding, Hett,y had done exactly that, but' then withdrew t,he case.) Alternatively, if t,he t,erms of a will do evidence a contract'ual obligation-as if A put's B's children t'hrough college, and B agrees to leave A his house in ret,urn-a situat'ion may arise in which a testat'or is not free unilat'erally t'o change t'he t'erms. I t was Het'ty Robinson's claim, supported with the 1862 will and other evidence, that she and her aunt had contracted to make mutual wills, specifically excluding Hetty's father Edward Robinson, so that in no case could he inherit the Howland fortune. However, Massachusetts law a t the time treated a beneficiary under a will as disqualified on account of self-interest from testifying about the circumstances surrounding the will, unless 503 called to testify by the opposite party or by the court.5 Without Hetty's testimony, the evidence was judged insufficient to support her claim that the 1862 will was in fulfillment of a contract. On this ground the court found for the defendant, Thomas Mandell, and it can only be conjectured whether it might have ruled otherwise had it considered the contested signatures to be valid. 4.1 Did She or Didn't She? So ends the case-without a finding on the question of most interest to the outsider. Clearly if the contested signatures were forgeries, Hetty Robinson was responsible for them. Did she trace her aunt's signature or didn't she? Here the amateur finds himself on shifting sands. Peirce includes in his testimony some further bits of evidence that point toward forgery, such as the position of the signature on the page, and the level of each letter. Osborn, a specialist in the field of questioned documents and a student of this case in particular, had no doubt whatever that the codicil signature was a forgery by tracing. However, he makes no comment on the J.Q. Adams signatures that appear to defy the usual tests of forgery. Examination of those of the 42 comparison signatures still extant shows a great deal of regularity, but one does not find the same sort of identity as between # 1 and # 10. But the original and both the contested signatures were said to have been written on the same day, and thus might well be more alike than signatures written many days apart. The tendency of all three of the signatures a t issue to follow one line from S through w, and then slope off for the last three letters in a slightly different direction, looks a t first compelling-until we notice that precisely this pattern is characteristic of most of the 42 comparison signatures. This is, perhaps, as far as a n amateur can go, and no unequivocal conclusion seems possible. 5. EPILOGUE Hetty Robinson Green-she had recently marriedlost this case, and other legal actions subsequently. However, far from being weak in those skills necessary to the management of so large a sum-the ostensible reason for putting the money in trust in the first placeHetty Green turned out to be a dynamic and effective operator in finance. According to contemporary newspaper accounts she devoted herself to the multiplication of her fortune and became one of the most colorful figures Wall Street had known for many years. Despite her enormous wealth, she lived for many years in a n inexpensive flat in Hoboken and took the ferry t o work in New York City each mprning. All this was part of a consciously projected image of penury, which even extended to her wearing old and worn clothing. Viewed as eccentric by many, she had a complex personality The American Law Review (1870, Vol. 4, pp. cusses this aspect of the case in considerable detail. 656-663), dis- 504 Journal of the American Statistical Association, September 1980 Evidence a t the trial was introduced suggesting that relations between Hetty and her aunt were by no means as cordial as Hetty made out, thus casting doubt on the credibility of Hetty's testimony. For further details of the ...

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Rutgers - BEN-ISRAEL - 711
Published: October 6, 1866 Copyright The New York Times
Rutgers - BEN-ISRAEL - 711
Sally ClarkPage 102/10/2007Using Statistical Evidence in Courts: A Case Study Or What Went Wrong in the Case of Sally Clark?1 IntroductionThe debate about the usage of statistical evidence in criminal courts has a long history.1 Whilst the ca
Rutgers - BEN-ISRAEL - 711
Airline Yield Management with Overbooking, Cancellations, and No-ShowsJANAKIRAM SUBRAMANIANIntegral Development Corporation, 301 University Avenue, Suite 200, Palo Alto, California 94301SHALER STIDHAM JR.Department of Operations Research, CB 318
Rutgers - BEN-ISRAEL - 711
Dynamic Pricing in Airline Seat Management for Flights with Multiple Flight LegsPENG-SHENG YOUDepartment of Business Administration, Chang Jung University, 396, Section 1, Chang Jung Road, Town of Kway Jen, Tainan 711, TaiwanConsider a multiple b
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Operations Management (33:623:386:03 & 10), Spring 2008Schedule (tentative)Class # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 -16 17 18 19 20 21 22 23 24 25 26 27 28 Date Wed Mon Wed Mon Wed Mon Wed Mon Wed Mon Wed Mon Wed Mon Wed Mon Wed Mon Wed Mon Wed
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A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69BCDEFGHLOOKUP, VLOOKUP and HLOOKUP
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Example of errors in computingThe numbers in Cells Ci are called Ci Let: C2 := 2*C1 - 1*C1 C3 := 3*C2 - 2*C1 C4 := 4*C3 - 3*C1, etc. All these should be equal to C1 This happens if C1 is an integer See what happens if: Example 1: C1 = 0.001 Example
Rutgers - BEN-ISRAEL - 386
CONTRACTORS3 contractors can do a job. The job has to be done in The relevant data is as follows: Contractor Can do the job in Charges 1 20 100 2 10 200 3 18 days 90 $/day 12 daysDetermine a plan to do the job in minimal cost. Contractors can be h
Rutgers - BEN-ISRAEL - 386
The Bags Problem: The problem data Time Required Hours/Unit Regular Deluxe Cutting 0.70 1.00 Sewing 0.50 0.83 Finishing 1.00 0.67 Inspection 0.10 0.25 Profit 10.00 9.00 Hours Available 630 600 708 135The Bags Problem: Data and variables Time Requir
Rutgers - BEN-ISRAEL - 386
SHELLEquipment Bag Problem - First Example LP Regular 0.7 0.5 1 0.1 $10.00 Regular 500 Deluxe vailable A 1 630 0.833 600 0.667 708 0.25 135 $9.00 Deluxe 300Cut/Dye Sew Finish Inspect/Pack ProfitVariable x1 Objective FunctionVariable x2Make
Rutgers - BEN-ISRAEL - 386
1100 1050 1000 950 900 850 800 750 700 650 600 550 500 450 400 350 300 250 200 150 100 50 0 -50 -100 -150 -200 Row 11100 1050 1000 950 900 850 800 750 700 650 600 550 500 450 400 350 300 250 200 150 100 50 0 -50 -100 -150 -200 Row 11100 1050 1000
Rutgers - BEN-ISRAEL - 386
11666531.xlsA 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16BCDEFGDiet Problem Corn 10 50 30 900 $0.60 0.00 Provided 46.54 243.65 100.00 4500.00 Soy 9 45 90 1200 $0.30 0.71 Oats 11 58 10 1000 $0.35 3.65 Cost $1.49 Fish Min Max 8 40 50 120 5 2
Rutgers - BEN-ISRAEL - 386
The Bike Wheels Problem: The dataSpokes Rims Wheels Machine time 0.04 0.6 Labor time 0.01 0.5 1.2 Cost 0.05 4 11 Selling price 0.1 10 50 Available 378 267Spokes Rims Wheels1 1-72 -1 1The Bike Wheels Problem: VariablesSpokes Rims Wheels Mach
Rutgers - BEN-ISRAEL - 386
11666619.xls1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22A B C D Bicycle Wheel Production Problem - "Grid" Version Activities Make Rims 0.6 0.5 0 1 0 $4.00 150EFMachine Time Labor Spokes Rims Wheels Direct Unit Cost Amount of Act
Rutgers - BEN-ISRAEL - 386
The Perfume Problem: The DataProcess Raw Material 1 1 3 0 4 0 $3.00 Activities Refine Brute 0 3 -1 1 0 0 $4.00 Refine Chanelle 0 2 0 0 -1 1 $4.00Raw Material Lab Time Regular Brute Luxury Brute Regular Chanelle Luxury Chanelle Direct Unit CostLi
Rutgers - BEN-ISRAEL - 386
11666542.xls1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25A B Perfume Problem - "Grid" VersionCDEFRaw Material Lab Time Regular Brute Luxury Brute Regular Chanelle Luxury Chanelle Direct Unit Cost Amount of Activity
Rutgers - BEN-ISRAEL - 386
PIGSKIN.XLSA 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20BCDEFMultiperiod Production Problem Start inventory 50 1 Demand Prod. cost/unit Holding cost/unit 100 $12.50 $0.625 2 150 $12.55 $0.628 Month 3 4 300 350 $12.70 $12.80 $0.63
Rutgers - BEN-ISRAEL - 386
11666599.xlsA 1 2 3 4 5 6 7 8BCDEFGHIndustrial Gases Transportation Problem Unit shipping costs to Customer Total 1 2 3 4 5 Available Plant 1 $8 $6 $7 $10 $9 45 FromPlant 2 $9 $12 $5 $13 $7 60 Plant 3 $14 $9 $12 $16 $5 55 Total requ
Rutgers - BEN-ISRAEL - 386
11666564.xlsA 1 2 3 4 5 6 7 8 9 10 11 12BCDGroovy Juice Mixers, Inc. Minimum % Tropical Breeze Guava Jive Maximum % Tropical Breeze Guava Jive Grape Guava Papaya 0% 20% 20% 0% 40% 0% Grape Guava Papaya 100% 25% 25% 100% 50% 5% Grape Guava P
Rutgers - BEN-ISRAEL - 386
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 Profit summary 28 29 30 31 Total profitBCDEFChandler Blending Problem Monetary inputsQuality level per barrel of crudesRequired quality level per barrel of prod
Rutgers - BEN-ISRAEL - 386
11666649.xlsA 1 2 3 4 5 6 7 8BCDPickles - Separate Advertising Min demand 5000 4000 30% 60% Adv Rate 3 5 Prod Cost $0.60 $0.85 Budget $16,000Sweet Dill Min Sweet Max SweetDATA11666649.xlsE 1 2 3 4 5 6 7 8FSelling $1.45 $1.75Co
Rutgers - BEN-ISRAEL - 386
INVEST.XLSA 1 2 3 4 5 6 7 8 9BCDEFGInvestment Problem Investment Min % Max% A 0% 30% B 25% 100% C 0% 40% Initial Cash Interest Rate Now $(1.00) $(1.00) Year 1 $0.20 $0.10 $(1.00) Year 2 $1.40 $1.60 Year 3 $1.25$1,000 8.00%DATAIN
Rutgers - BEN-ISRAEL - 386
INVEST.XLSA 1 2 3 4 5 6 7 8 9 10BCDInvestment Problem: Data Interest Rate Investment A B C Funds 8% Now $1.00 $1.00 $0.00 $1,000.00 8% Year 1 -$0.20 -$0.10 $1.00 $0.00 8% Year 2 $0.00 -$1.40 $0.00 $0.00DATAINVEST.XLSE 1 2 3 4 5 6 7 8
Rutgers - BEN-ISRAEL - 386
conmine1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28A B Consolidated MiningCDEFGFrom Mine Blue Mesa Dry PassShipping Cost To Boise West TX Capacity $4.50 $3.00 800 $3.50 $6.00 1000 Boise West TX $17.00 $
Rutgers - BEN-ISRAEL - 386
11666586.xls1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41A B C National Vehicular Seating, Inc. Production Data Assembly Hours Sewing Hours Weight Cost, Plant 1 Cost, Plant 2 Reg
Rutgers - BEN-ISRAEL - 386
11666663.xlsA 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23BCDEFWidgetCo Project Scheduling (AON) Requires C -1 -C -26 -26Code A B C D E F G Code A B C D E F GActivity Train Workers Purchase RM Make SA 1 Make SA 2 Insp
Rutgers - BEN-ISRAEL - 386
11666670.xlsA 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22BCDEWidgetCo Project Scheduling (AON)-SIMPLE vRequiresCode A B C D E F G Code A B C D E F GActivity Train Workers Purchase RM Make SA 1 Make SA 2 Inspect SA 2 Asse
Rutgers - BEN-ISRAEL - 386
Rutgers - BEN-ISRAEL - 386
11527167.xls1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22A B C D WidgetCo Project with Crashing (AON)E F Deadline 25GHIJKCode A B C D E FActivity Train Workers Purchase RM Make SA 1 Make SA 2 Inspect SA 2 AssembleDura
Rutgers - BEN-ISRAEL - 386
11666606.xlsABCDEFG1 WidgetCo Project with Crashing Deadline 2 3 Cost/Day Max Days 4 Code Activity Duration to Crash Crash 5 A Train Workers 6 $10.00 5 6 B Purchase RM 9 $20.00 5 7 C Make SA 1 8 $3.00 5 8 D Make SA 2 7 $30.00 5 9 E In
Rutgers - BEN-ISRAEL - 386
11666540.xlsA 1 2 3 4 5 6 7 8 9 10 11BCDEFGStockco Capital Budgeting Problem1 1 $19,000 $7,000 $50,000 Investment 2 1 $21,000 $9,000 3 0 4 1 $10,000 Total cost $5,000 $21,000 <=Investment level Total Net Return Investment cost Tota
Rutgers - BEN-ISRAEL - 386
ValuesA 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21BCDEPlane Loading Problem Item Weight 1 4,000 2 800 3 2,000 4 1,500 Capacity 30,000 Cost/Unit $0.05 Item 1 2 3 4 Take 3 10 1 5 Weight 29,500 Weight $1,475Alternative Ship Volu
Rutgers - BEN-ISRAEL - 386
11527132.xlsA 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22BCDEFGHIMachinco Assignment of Jobs to Machines Problem Costs to perform jobs on various machines Job 1 2 3 4 Machine 1 14 5 8 7 2 2 12 6 5 3 7 8 3 9 4 2 4 6 10
Rutgers - BEN-ISRAEL - 386
11666672.xlsA 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28BCDEFGHIJKLMAn assignment problem with 8 persons and 8 jobs Costs of assinging persons to jobs 1 2 1 6 2 11 6 3 12 2 1 4 10 7 12 1 3 1
Rutgers - BEN-ISRAEL - 386
milkemA 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24BCDEFGProblem 5.4 - "Boris Milkem" Data on selling prices of assets (in $millions) Asset 1 Asset 2 Asset 3 Asset 4 Asset 5 Asset 6 Sold in year 1 15 16 22 10 17 19
Rutgers - BEN-ISRAEL - 386
11527124.xlsA 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17BCDEFGHIJKLMNWestern Airlines Set Covering ProblemCities Potential hub Covered AT BO CH DE HO LA NO NY AT 1 0 1 0 1 0 1 1 BO 0 1 0 0 0 0 0 1 CH 1 0 1 0 0 0 1 1 DE 0
Rutgers - BEN-ISRAEL - 386
A B C D E 1 Crew Assignment Problem: SmallTime Airlines 2 3 Duty Plan Cost 101 102 103 4 5 6 7 8 9 10 11 12 13 14 15 16 Covered: 17 18 Total Cost 19FGHIJKLMNOP201Flights Covered 202 203 401 402403501502503Use?A B
Rutgers - BEN-ISRAEL - 386
11666654.xlsA 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16BCDEProblem 5.5 Data on pitchers Righty? (1 if yes, 0 if not) Cost (in $millions) Victories added Pitchers to sign (1 if signed, 0 if not) RS 1 Budget constraint (in $millions) Spent $11
Rutgers - BEN-ISRAEL - 386
ValuesA 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31BCDGandy Fixed Charge Clothing ProblemInput data Shirts 3 4 40 $12 $6 $200 Product Shorts 2 3 53.33 $8 $4 $150 Pants 6 4 25 $15 $8 $100Labor hours
Rutgers - BEN-ISRAEL - 386
11666537.xlsA 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25BCDPower Plant Problem (5.17) Boiler Min Steam 1 500 2 300 3 400 Turbine Min Steam 1 300 2 500 3 600 Run Boiler? 1 2 3 Total: Run Turbine? 1 2 3 0 1 1 0 1 1 Max
Rutgers - BEN-ISRAEL - 386
11666659.xlsA 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26BCDEFGHIJKLMNOHiring Students Min Shifts Initials CM FI GR HS JD JE SR TR Availability Mon Tue Wed Thu Fri Profit/Shift AM PM AM PM AM P
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ValuesA 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46BCDEFGHIJKHuntco Plant/Warehouse Location Problem Plant to warehouse unit production, ship
Rutgers - BEN-ISRAEL - 386
LOGICAL CONSTRAINTS AND BINARY VARIABLESADI BEN-ISRAELLet A, B, C, denote some actions, and let A , B , C , be the corresponding decisions, i.e., X = 1, if action X is taken; 0, otherwise.We express logical conditions on the actions by con
Rutgers - BEN-ISRAEL - 386
Buying widgetsThe XYZ Corp. has a budget of $500,000, which it intends to use for buying widgets. There are three vendors, each with unlimited supply, who offered the following terms. Vendor 1 stipulates that an order must be no less than 4,000 widg
Rutgers - BEN-ISRAEL - 386
Tossing a CoinA coin shows "1" and "0" with probabilities p, 1-pp 0.4 Value Probability 0 0.6 1 0.4Simulated value #Rolling a dieA die shows "1", , "6" with given probabilitiesValue Probability 1 0.166667 2 0.166667 3 0.166667 4 0.166667 5 0
Rutgers - BEN-ISRAEL - 386
A Guessing GameThrow two dice and guess a number from 2 to 12 Your opponent always chooses the number in cell C10 If your number is closer to to the sum of the dice you win 1 point. If your opponent is closer you w There are 11 scenarios. Find the
Rutgers - BEN-ISRAEL - 386
11666634.xlsTranslator Hiring Problem Chance of Overtime Availability Fixed Cost per Translator Regular Order Cost Overtime Order Cost Revenue per Filled Order Translators Hired Actual Demand Overtime Translators Available Regular Orders Filled Ove
Rutgers - BEN-ISRAEL - 386
11666639.xlsA 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25BCDETranslator Hiring Problem Chance of Overtime Availability Fixed Cost per Translator Regular Order Cost Overtime Order Cost Revenue per Filled Order Transla
Rutgers - BEN-ISRAEL - 386
0.25 0.24 0.23 0.22 0.21 0.2 0.19 0.18 0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20200.25 0.24 0.23 0.22 0.21 0.2 0.19 0.18 0.17 0.16 0.15 0.14 0.13 0.1
Rutgers - BEN-ISRAEL - 386
11666627.xlsA 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19BCDEOverbooking ModelTicket Price Penalty Seats Expected Demand Probability of Coming Overbooking Demand Tickets Sold Passengers Arrived Passengers Seated Passengers Not Seated
Rutgers - BEN-ISRAEL - 386
An illustration of the Central Limit TheoremWe generate 100 independent RV's in D17:D116, simulate their sum in D118, and compare its distribution with the "corresponding" normal distribution. Each RV is uniformly distributed on [0,a] where the a's
Rutgers - BEN-ISRAEL - 386
11666600.xls1 2 3 4 5 6 7 8 9 10 11 12 13 14 15A Classic 3-Door "Paradox" Door with car Door we pick at first Door Has car Is the door we picked Is a door Monty can show Random value Shown by Monty Door we would switch to Get car if don't switch
Rutgers - BEN-ISRAEL - 386
PROJECT DURATIONA project has 8 activities, see diagram below. The activity durations are random, with triangular distribution and Min, Mode, Max as given. Simulate: (a) The project duration (b) For each path, the probability that it is critical.A
Rutgers - BEN-ISRAEL - 386
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25BCDEPower SupplyCost per kWh Selling price per kWh Startup Cost ($1000's) Capacity (MW) Expected Municipal Demand (MW) Standard Deviation of Municipal Demand (MW) Minimum I
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17A Insurance Reserve Capital ProblemBCDE1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17A B Insurance Reserve Capital ProblemCDEYASAI Simulation Output Workbook Sheet Start Date Start Time Run Ti
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11527204.xls1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18A B Inventory Simulation Mean demand Fixed order cost Unit cost Sales price Holding cost Salvage value Beginning inventory Reorder point Reorder quantityCDEFGHIJ400 $600 $1
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11666558.xls1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82
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11527142.xlsA B C D E F 1 Westland Wranglers 2 Average Horses Captured per Day 4 3 Average Sales Demand per Day 4.1 4 Sales Price Per Horse $150.00 5 "Salvage" Value of Horses in Corral at End $130.00 6 Cost per Day of Keeping Horse in Corral $8.00
Rutgers - BEN-ISRAEL - 386
11666561.xlsA B C D E F G H I J 1 Truck Service Facility Expansion 2 Average arrival rate 3.8 Daily Operation Cost $425 per occupied bay per day 3 4 Number of Bays #NAME? 5 6 7 8 5 Incremental Daily Cost #NAME? $175 $325 $475 6 7 First Second Third
Rutgers - BEN-ISRAEL - 386
The St Petersburg ParadoxTwo players A & B, and a coin The coin gives H with probability in Cell B12 The coin is tossed until first H appears If this happens in Trial n, B pays A the amount 2^(n-1) To make the game fair, A pays B the value of the ga
Rutgers - BEN-ISRAEL - 386
BUFFON'S NEEDLE PROBLEMA needle of length L = #NAME? is dropped on equally spaced parallel lines a distance D = 1 apart. The probability that the needle will cross a line is 2 L / Pi D This formula makes sense fo L< Pi D/2 for otherwise it gives val