12. Van Dantzig's collective marks. Let X; be the number of balls in the i th of a sequence of bins, and assume the X; are independent with common probability generating function Gx. There are N such bins, where N is independent of the X; with probability generating function Gn. Each ball is unmarked with probability u, and marked otherwise, with marks appearing independently. (a) Show that the probability , that all the balls in the i th bin are unmarked, satisfies n = Gx(u). (b) Deduce that the probability that all the balls are unmarked equals Gn(Gx(u)), and deduce the random sum formula of Theorem (5.1.25). (c) Find the mean and variance of the total number T of balls in terms of the moments of X and N, and compare the argument with the method . (d) Find the probability generating function of the total number U of unmarked balls, and deduce the mean and variance of U.