The probability W(n) that an event characterized by a probability p occurs n times in N...

  

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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 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." 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."