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07-Poisson Distribution

# 07-Poisson Distribution - IE 111 Fall Semester 2009 Notes...

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IE 111 Fall Semester 2009 Notes distributed 10/16/09 The Poisson Distribution The Poisson Distribution is another extremely important distribution in Probability and Statistics. It is especially important for Industrial Engineers for reasons that will become apparent. The distribution was discovered by Siméon-Denis Poisson (1781 –1840 ) and published, together with his probability theory, in 1838 in his work Recherches sur la probabilité des jugements en matières criminelles et matière civile ("Research on the Probability of Judgments in Criminal and Civil Matters"). The Poisson distribution describes the probability that a random event will occur in a time or space interval under the conditions that the probability of the event occurring is very small, but the number of trials is very large so that the event actually occurs a few times. The word “Poisson” is a French word, thus is pronounced differently depending on the ability to speak French. Common pronunciations include: 1. Poy saan 2. Pwah saan 3. Pwah son Since 1. is really a butchery of French and 3. requires a French accent thereby appearing snooty, I usually use 2. The Poisson distribution is derived from the Binomial by letting N →∞ and p 0 but all the time keeping Np= λ constant. For example: N 10 100 1000 10,000 ----------------------------------------- ....... λ = 1 always p 0.1 0.01 0.0001 0.00001 In the limit, the Binomial approaches the Poisson (see book for details). The Poisson PMF is given by: P X (x) = e - λ λ x / x! for x in {0,1,2,. ....} Note that the domain includes all non-negative integers, and that the Poisson distribution has a single parameter λ . 1

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The mean and variance of the Poisson can be shown (after a lot of tedious math) to be: E(X) = λ and V(X) = λ λ = = = - = - = = = = - = - - = - = - = - ) 1 ( ! ) ( ! ) 1 ( ! ) 1 ( ! ! ) ( 0 1 1 1 1 0 k k j j j j j j j j k e j e j e j je j je X E Derivation of the variance is similar, and is in the book. The fact that the mean and variance are equal is an unusual, but not terribly useful result. One use of the Poisson distribution is as an approximation to the Binomial. As we have seen, one difficulty of using the Binomial is that it is quite time consuming to calculate the CDF in order to evaluate P(X x). Example Suppose 100 chips are manufactured on a line which has historically produced 5% defective chips. What is the probability that 5 or fewer of the 100 are defective? Letting X = the number of defective chips, it should be clear by now that X is a Binomial random variable with N=100 and p=0.05. We want to find P(X 5). Using the Binomial, P(X 5) = P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)+P(X=5) = 100 C 0 0.05 0 (0.95) 100 + 100 C 1 0.05 1 (0.95) 99 + 100 C 2 0.05 2 (0.95) 98 + 100 C 3 0.05 3 (0.95) 97 + 100 C 4 0.05 4 (0.95) 96 + 100 C 5 0.05 5 (0.95) 95 = 0.0059205 + 0.0311606 + 0.0811817 + 0.1395756 + 0.1781426 + 0.1800178 = 0.6159988
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07-Poisson Distribution - IE 111 Fall Semester 2009 Notes...

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