Question: 5. A branching process (Xn : n 0) has P(X0 = 1) = 1. Let the total number of individuals in the first n

5. A branching process (Xn : n ≥ 0) has P(X0 = 1) = 1. Let the total number of individuals in the first n generations of the process be Zn, with probability generating function Qn. Prove that, for n ≥ 2, Qn(s) = sP1(Qn−1(s)), where P1 is the probability generating function of the family-size distribution. (Oxford 1974F)

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