Suppose that events occur in accordance with a Poisson process with rate A, and that an event occurring at time s, independent of the past, contributes a random amount having distribution F,, s 0. Show that W, the sum of all contributions by time t, is a compound Poisson random variable That is, show that W has the same distribution as X,, where the X, are independent and identically distributed random variables and are independent of N, a Poisson random variable Identify the distribution of the X, and the mean of N