Probability | Part IA, 2006

Suppose that a population evolves in generations. Let ZnZ_{n} be the number of members in the nn-th generation and Z01Z_{0} \equiv 1. Each member of the nn-th 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 with the same 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. Use this to compute the variance of ZnZ_{n}.

Now consider the total number of individuals in the first nn generations; this number is a random variable and we write HnH_{n} for its generating function. Find a formula that expresses Hn+1(s)H_{n+1}(s) in terms of Hn(s),G(s)H_{n}(s), G(s) and ss.

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