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Unformatted text preview: Poisson Distribution (parameter is lamda, ) To introduce the Poisson, we will start with it as an approximation to the Binomial. Fact from calculus: t e t + 1 when t . Consider the Binomial Distribution when n is large a nd p is small (close to 0). np n p n n X e e p p p n p = = = ) ( ) 1 ( ) 1 ( ) ( For another k, where k is a positive (but small compared to n), then np k k j np X X X k n k X e k np e j np p p k p p p k n k p = = = = ! ) ( * ) ( ) ( * ) ( ) ( ) 1 ( ) ( 1 (equation 1) Lastly, when k is not small relative to n, both sides are roughly 0. It is not surprising given the preceding statement, that the right side of these equations are both poisson distributions where =np (and x is 0 or k respectively). From this approximation we can gather that poisson is used for what we call rare events (small p). This is often useful in disease and mortality problems. is often useful in disease and mortality problems....
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