elemprob-page13

# elemprob-page13 - so E X 2 = E X 2 X | EX = Î 2 Î and hence...

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Unformatted text preview: so E X 2 = E ( X 2- X ) + | EX = Î» 2 + Î» , and hence Var X = Î» . Example. Suppose on average there are 5 homicides per month in a given city. What is the probability there will be at most 1 in a certain month? Answer. If X is the number of homicides, we are given that E X = 5. Since the expectation for a Poisson is Î» , then Î» = 5. Therefore P ( X = 0) + P ( X = 1) = e- 5 + 5 e- 5 . Example. Suppose on average there is one large earthquake per year in California. Whatâ€™s the probability that next year there will be exactly 2 large earthquakes? Answer. Î» = E X = 1, so P ( X = 2) = e- 1 ( 1 2 ). We have the following proposition. Proposition 5.1. If X n is binomial with parameters n and p n and np n â†’ Î» , then P ( X n = i ) â†’ P ( Y = i ) , where Y is Poisson with parameter Î» . The above proposition shows that the Poisson distribution models binomials when the probability of a success is small. The number of misprints on a page, the number of automobile accidents, the number ofa success is small....
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