CEE_304__LectureExtra_14

CEE_304__LectureExtra_14 - COpyright © 1993 Grolier...

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Unformatted text preview: COpyright © 1993, Grolier Electronic Publishing, inc. All rights reserved. . Poisson distribution The Poisson distribution, named for the French mathematician Simeon D. Poisson (1781—1840), is a PROBABILITY model. It describes the probability that a random event will occur in a time or space interval under certain conditions: that the probability of the event occurring is very small, but that the number of trials--time or space intervals—-is very large so that the event actually occurs a few times. The Poisson distribution describes, for example, the number of misprints per page in a book, or the number of automobile deaths per month in a large city. The Poisson probability, P, that the count is k is arrived at by using the following division: (e to the -m power times In to the k power) divided by kl, where m is the average number of occurrences of the event in the time or space interval, k is a positive integer, and k! = k(k — l)(l< - 2) . . . 1. The Poisson distribution is related to the BINOMIAL DISTRIBUTION and the NORMAL DISTRIBUTION. Bibliography: Feller, William, An Introduction to Probability Theory and Its Applications, vol. 1, 3d ed. (1968). Addicts in high places, how to order seafood, the disadvantage advantage, and other matters. port from onahue and Op h. ASK MR. STATISTICS I Dear Oddsgiver.‘ I am in the seafood distri- bution business and find myself endlessly wrangling with supermarkets about appropri- ate order sizes, especially with high-end tidbit products like our matjes herring in super- spiced wine, which we let them have for $4.25, and still they take only a half-dozen jars, thereby running the risk of getting sold out early in the week and causing the better class of customers to storm out empty—hand- ed. How do I get them to realize that lowball- ing on inventories is usually bad business, also to at least try a few jars of our pickled crappie balls? —HEADED FOR A BREAKDOWN Dear Picklehead: The science of statis- tics has much to offer people puzzled by seafood inventory problems. Your salva- tion lies in the Poisson distribution, “pois- son” being French for fish and, of arguably greater relevance, the surname of a 19th-century French probabilist. Siméon Poisson’s contribution was to de— velop a method for calculating the likeli- hood that a specified number of successes will occur given that (a) the probability of success on any one trial is very low but (b) the number of trials is very high. A real- world example often mentioned in the liter- ature concerns the distribution of Prussian cavalry deaths from getting kicked by hors- ’ es in the period 1875—94. As you would expect of Teutons, the L Prussian military kept meticulous records on horse-kick deaths in each of its army corps, and the data are neatly summarized in a 1963 book called Lady Luck, by the late Warren Weaver. There were a total of 196 kicking deaths—these being the, er, “suc— cesses.” The “trials” were each army corps’ observations on the number of kicking deaths sustained in the year. So with 14 army corps and data for 20 years, there were 280 trials. We shall not detain you with the Poisson formula, but it predicts, for example, that there will be 34.1 in- stances of a corps’ having exactly two deaths in a year. In fact, there were 32 such cases. Pretty good, eh? Back to seafood. The Poisson calculation is appropriate to your case, since the likeli- hood of any one customer’s buying your overspiced herring is extremely small, but the number of trials—i.e., customers in the store during a typical week—is very large. Let us say that one customer in 1,000 deigns to buy the herring, and 6,000 cus- tomers visit the store in a week. So six jars are sold in an average week. But the store manager doesn’t care about average weeks. What he’s worried about is having too much or not enough. He needs to know the probabilities assigned to dilTerent sales levels. Our Poisson distribution shows the following morning line: The chance of fewer than three sales—only 6.2%. Of four to six sales: 45.5%. Chance of losing some sales if the store elects to start the week with six jars because that happens to be the aver- age: 39.4%. If the store wants to be 90% sure of not losing sales, it needs to start with nine jars. There is no known solution to the prob- lem of pickled crappie balls. ‘ MARCH7, 1994 FORTUNE 133 — 50:33- 1 »/— @WDv‘ZL‘] } __. I ’79-;' ' ‘A‘é I *‘ \fl M [Bo/«J and»; I #1 Extesion: The variable t need not correspond to time. It could correspond to distance between defects in a cable or beam; it could be distance along a road between events. It could correspond to ' different boxes in a two- or three-dimensional space; for these two or three dimensional problems Tk wouldn't make sense. Lecture 14; page 9 ...
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This note was uploaded on 10/06/2010 for the course CEE 3040 taught by Professor Stedinger during the Fall '08 term at Cornell.

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CEE_304__LectureExtra_14 - COpyright © 1993 Grolier...

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