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l15_poisson

# l15_poisson - Poisson Distribution Reading Ross Ch 4 Sec 7...

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04/01/2009 CS206 - Intro. to Discrete Structures II 1 Poisson Distribution Reading: Ross, Ch 4., Sec. 7. Wednesday, April 1, 2009

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04/01/2009 CS206 - Intro. to Discrete Structures II 2 Example 1 Greedy counterfeiter problem: The king’s minter boxes n coins to a box. Each box contains m false coins. The king suspects the minter and randomly draws 1 coin from each of n boxes and has them tested. What is the chance that the sample of n coins contains exactly r false ones? X - find a false coin in a box, X ~ Bernoulli, w/p p = m/n S n - # false coins in n boxes, S n ~ Bi(n,p) P(S n =r) = C(n,r) (m/n) r (1-m/n) n-r Wednesday, April 1, 2009
04/01/2009 CS206 - Intro. to Discrete Structures II 3 Example 1 (cont’d) What happens as n becomes large, while m & r are fixed? P(S n =r) = n! / [ (n-r)! r! ] (m/n) r (1-m/n) n-r = 1/r! n(n-1)(n-2)…(n-r+1)/n r m r (1-m/n) n (1-m/n) -r lim n !1 P(S n =r) = 1/r! 1 m r e -m 1 = m r /r! e -m because lim n !1 (1-m/n) n = e -m lim n !1 (1-m/n) -r = 1 lim n !1 n(n-1)(n-2)…(n-r+1)/n r = 1 Wednesday, April 1, 2009

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04/01/2009 CS206 - Intro. to Discrete Structures II 4 Poisson Random Variable Random variable X {0,1,…} is said to have a Poisson distribution if P(X=k) = λ k /k! e - λ with parameter λ >0 . Siméon D. Poisson 1781-1840 Wednesday, April 1, 2009
04/01/2009 CS206 - Intro. to Discrete Structures II 5 Examples of Poisson RV Wide range of applications: # misprints on a book page # customers entering a post office on a give day # atomic particles discharged in a fixed time from radioactive material # vehicles entering an intersection in a fixed time # contaminants in a unit of semiconductor material

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