0 1. Consider a branching process with Z = 1 and let Y = Z+Z+...+Z, be the total progeny. Derive the pgf R,(s) of the total progeny and hence find its i) Mean ii) Variance (when #1) 2. Calculate the PGF of the second generation. i) ii) distribution of the second distribution iii) probability of extinction by third generation iv) probability of extinction at third generation v) probability of ultimate extinction. for the following branching process with Z = 1 and branching mechanism a) p=0.1, p=0.7 and p=0.2. b) p = 0.3, p=0.4 and p, 0.3. P 1/6, p 1/2 and p, 1/3 3. In a branching process, the probability that any individual has j individuals is given by 1 Po =0, P 21 2 i) Show that the probability generating function of the first generation is given by G(s)= S 2-s ii) Determine the pgf of the second, third and fourth generation.