The probability W(n) that an event characterized by a probability p occurs n times in N trials was shown to be given by the binomial distribution N W(n) (1) ni(N n)1 P (1 p) Consider a situation where the probability p is small (p 1) and where one is interested in the case n N (Note that if N is large, W(n) becomes very small if n N because of the smallness of the factor p when p 1 Hence W(n) is indeed only appreciable when n N ) Several approximations can then be made to reduce (1) to simpler form (a) Using the result In (1 p) P, show that (I (b) Show that N (N n) N (c) Hence show that (1) reduces to p eN W(n) n (2) where A Np is the mean number of events The distribution (2) is called the Poisson distribution

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The probability W(n) that an event characterized by a probability p occurs n times in N trials was shown to be given by the

The probability W(n) that an event characterized by a probability p occurs n times in N trials was shown to be given by the binomial distribution N! W(n) (1) ni(N - n)1 P"(1 - p)- Consider a situation where the probability p is small (p

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