Probability | Part IA, 2002

(a) Explain what is meant by the term 'branching process'.

(b) Let XnX_{n} be the size of the nnth generation of a branching process in which each family size has probability generating function GG, and assume that X0=1X_{0}=1. Show that the probability generating function GnG_{n} of XnX_{n} satisfies Gn+1(s)=Gn(G(s))G_{n+1}(s)=G_{n}(G(s)) for n1n \geq 1.

(c) Show that G(s)=1α(1s)βG(s)=1-\alpha(1-s)^{\beta} is the probability generating function of a non-negative integer-valued random variable when α,β(0,1)\alpha, \beta \in(0,1), and find GnG_{n} explicitly when GG is thus given.

(d) Find the probability that Xn=0X_{n}=0, and show that it converges as nn \rightarrow \infty to 1α1/(1β)1-\alpha^{1 /(1-\beta)}. Explain carefully why this implies that the probability of ultimate extinction equals 1α1/(1β)1-\alpha^{1 /(1-\beta)}.

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