PROBLEMS 3 1. This problem considers the Poisson distribution, a probability distribution for a discrete random variable which was first used by Siméon-Denis Pois- son to describe seemingly random criminal events in Paris in 1837. If inde- pendent events have a constant tendency to occur and if the average rate of occurrence is a, then the probability that n events actually occur is given by * with n0,1,2.x. (a) By noting that show that thereby verifying that the Poisson distribution is normalized. (b) By using n/nt - 1/n – 1) and a - at, show that thereby verifying that the average rate of occurrence, or the expectation value (m), is equal to a. (e) By using similar techniques, find (w) and show, using Eq. (3.4), that the standard deviation of the Poisson distribution is given by An- Va.

PROBLEMS 3 1. This problem considers the Poisson distribution, a probability distribution for a discrete random variable which was first used by Siméon-Denis Pois- son to describe seemingly random criminal events in Paris in 1837. If inde- pendent events have a constant tendency to occur and if the average rate of occurrence is a, then the probability that n events actually occur is given by * with n0,1,2.x. (a) By noting that show that thereby verifying that the Poisson distribution is normalized. (b) By using n/nt - 1/n – 1) and a - at, show that thereby verifying that the average rate of occurrence, or the expectation value (m), is equal to a. (e) By using similar techniques, find (w) and show, using Eq. (3.4), that the standard deviation of the Poisson distribution is given by An- Va.