Let {Xn, n 0} be a branching process, where X, is the number of individuals born at time n, and X = 1. Suppose Z is the family size distribution, and Z ~ Geometric (p) such that P(Z = k) = pqk, k = 0, 1, 2, ... (a) Let G(s) = E[s] be the probability generating function on Z. Show that Show that and state the radius of convergence. (b) Use G(s) to find E[Z] when Z ~ Geometric (p). (c) Suppose that Z ~ Geometric (p =) (i) G(s) = (v) Now, suppose that Z~ extinction? 1- qs G(s) = (ii) Let G (s) be the probability generating function of X. Write down an expression for G (s). (iii) Using your answers in previous parts, find P(X; = 0), i = 1, 2, 3, 4. (iv) Find the probability of eventual extinction. 2 5 - 3s Poisson (2 = ). What is the probability of eventual