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poisson_notes2 - Some notes on the Poisson distribution...

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Some notes on the Poisson distribution Ernie Croot October 7, 2010 1 Introduction The Poisson distribution is one of the most important that we will encounter in this course – it is right up there with the normal distribution. Yet, because of time limitations, and due to the fact that its true applications are quite technical for a course such as ours, we will not have the time to discuss them. The purpose of this note is not to address these applications, but rather to provide source material for some things I presented in my lectures. 2 Basic properties Recall that X is a Poisson random variable with parameter λ if it takes on the values 0 , 1 , 2 , ... according to the probability distribution p ( x ) = P ( X = x ) = e - λ λ x x ! . By convention, 0! = 1. Let us verify that this is indeed a legal probability density function (or “mass function” as your book likes to say) by showing that the sum of p ( n ) over all n 0, is 1. We have summationdisplay n =0 p ( n ) = e - λ summationdisplay n =0 λ n n ! . This last sum clearly is the power series formula for e λ , so summationdisplay n =0 p ( n ) = e - λ e λ = 1 , 1
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as claimed.
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