2.II.9F
Suppose that a population evolves in generations. Let be the number of members in the -th generation and . Each member of the -th generation gives birth to a family, possibly empty, of members of the -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 .
Let be the generating function of . State and prove a formula for in terms of . Use this to compute the variance of .
Now consider the total number of individuals in the first generations; this number is a random variable and we write for its generating function. Find a formula that expresses in terms of and .
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