Paper 2, Section II, F

Probability | Part IA, 2010

In a branching process every individual has probability pkp_{k} of producing exactly kk offspring, k=0,1,k=0,1, \ldots, and the individuals of each generation produce offspring independently of each other and of individuals in preceding generations. Let XnX_{n} represent the size of the nnth generation. Assume that X0=1X_{0}=1 and p0>0p_{0}>0 and let Fn(s)F_{n}(s) be the generating function of XnX_{n}. Thus

F1(s)=EsX1=k=0pksk,s1F_{1}(s)=\mathbb{E} s^{X_{1}}=\sum_{k=0}^{\infty} p_{k} s^{k},|s| \leqslant 1

(a) Prove that


(b) State a result in terms of F1(s)F_{1}(s) about the probability of eventual extinction. [No proofs are required.]

(c) Suppose the probability that an individual leaves kk descendants in the next generation is pk=1/2k+1p_{k}=1 / 2^{k+1}, for k0k \geqslant 0. Show from the result you state in (b) that extinction is certain. Prove further that in this case

Fn(s)=n(n1)s(n+1)ns,n1F_{n}(s)=\frac{n-(n-1) s}{(n+1)-n s}, \quad n \geqslant 1

and deduce the probability that the nnth generation is empty.

Typos? Please submit corrections to this page on GitHub.