Consider a branching process for which the generator function for the branching probabilities is given by

(a) What is the probability that an individual produces k offspring?
(b) What is the average branching ratio?
(c) For which value of p is the process a critical branching process?
(d) Show that the generator function for the size of the progeny Yn after n generations,
g_Yn (s), is given by

where α = p(1 − p) and there are a total of n − 1 factors α in the nested fraction n on the right-hand side.

g(s) = 1-P 1-ps