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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 trialstime 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 highend tidbit
products like our matjes herring in super
spiced wine, which we let them have for
$4.25, and still they take only a halfdozen
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 19thcentury 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 horsekick 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 threedimensional 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.
 Fall '08
 Stedinger

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