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elemprob-fall2010-page17

elemprob-fall2010-page17 - so E X 2 = E(X 2 X E X = 2 and...

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so E X 2 = E ( X 2 - X ) + E X = λ 2 + λ , and hence Var X = λ . An 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 λ , then λ = 5. Therefore P ( X = 0) + P ( X = 1) = e - 5 + 5 e - 5 . An example. Suppose on average there is one large earthquake per year in California. What’s the probability that next year there will be exactly 2 large earthquakes? Answer: λ = E X = 1, so P ( X = 2) = e - 1 ( 1 2 ). We have the following proposition. Proposition 6.1 If X n is binomial with parameters n and p n and np n λ , then P ( X n = i ) P ( Y = i ) , where Y is Poisson with parameter λ . Proof. For simplicity, let us suppose λ = np n . In the general case we use λ n = np n . We write P ( X n = i ) = n ! i !( n - i )! p i
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