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poissnotes - Summary Notes: Poisson Distribution A random...

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Summary Notes: Poisson Distribution A random variable X is said to have a Poisson distribution if the probability mass function is: p ( x ; ¸ ) = e ¡ ¸ ¸ x x ! , for x = 0 ; 1 ; 2 ;::: and some ¸ > 0 . Note 1 The Poisson distribution is useful in modelling the number of events that occur in space or time, where the temporal or spatial unit is well-de…ned. Theorem 1 Suppose that in the binomial pmf b ( x ; n;p ) , we let n ! 1 and p ! 0 in such a way that np remains …xed at a value ¸ > 0 . Then b ( x ; n;p ) ! p ( x ; ¸ ) . Proof Let X be a binomial random variable with parameters n and p . Let ¸ = np . Then: P ( X = i ) = μ n i p i (1 ¡ p ) n ¡ i = n ! ( n ¡ i )! i ! μ ¸ n i μ 1 ¡ ¸ n n ¡ i = n ( n ¡ 1)( n ¡ 2) ¢¢¢ ( n ¡ i + 1) n i ¸ i i ! ¡ 1 ¡ ¸ n ¢ n ¡ 1 ¡ ¸ n ¢ i : Now for large n and appreciable ¸ , ¡ 1 ¡ ¸ n ¢ i is approximately 1, ¡ 1 ¡ ¸ n ¢ n is approximately e ¡ ¸ [from calculus we know that lim n !1 ¡ 1+ x n ¢ n = e x ; thus
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poissnotes - Summary Notes: Poisson Distribution A random...

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