Probability | Part IA, 2008

A population evolves in generations. Let ZnZ_{n} be the number of members in the nnth generation, with Z0=1Z_{0}=1. Each member of the nnth generation gives birth to a family, possibly empty, of members of the (n+1)(n+1) th generation; the size of this family is a random variable and we assume that the family sizes of all individuals form a collection of independent identically distributed random variables each with generating function GG.

Let GnG_{n} be the generating function of ZnZ_{n}. State and prove a formula for GnG_{n} in terms of GG. Determine the mean of ZnZ_{n} in terms of the mean of Z1Z_{1}.

Suppose that Z1Z_{1} has a Poisson distribution with mean λ\lambda. Find an expression for xn+1x_{n+1} in terms of xnx_{n}, where xn=P{Zn=0}x_{n}=\mathbb{P}\left\{Z_{n}=0\right\} is the probability that the population becomes extinct by the nnth generation.

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