Consider a general branching process. Let g(s) denote the generator function for the branching probabilities and let g_Zn (s) denote the generator function for the size Z_n of the nth generation. Furthermore, let x_n = Prob{extinction by generation number n}.

Equation (5.46)

(a) Show the relation gz(s) = g(gz-($)). -1 (b) Show that (c) Show that the probability x = lim xn = g(xn1). xn that the process eventually goes extinct is therefore given by the smallest root in s = g(s). Consider a branching process with the following branching probabilities: Po= q P = 0 P2P Pk = 0 k 2. (d) For which value of p is this process critical? (e) Determine how the average size of the nth generation depends on p. (5.45) (f) For arbitrary p = (0,1) determine the probability that the process eventually goes extinct. Now consider a branching process with branching probabilities given by Pk=qpkk = {0} UN. (5.46) (g) Let denote the average branching ratio, consider 1 - g(s) and show that for the process in Eq. (5.46) the generator function satisfies the following relation: