Introduction to Probability · Problem 10.2.2

Generating Functions — Grinstead & Snell

Mathematics — Probability · branching-processes

Problem

Let Z1,Z2,…,ZNZ_1, Z_2, \ldots, Z_N describe a branching process in which each parent has jj offspring with probability pjp_j. Find the probability dd that the process dies out if (a) p0=1/2p_0 = 1/2, p1=p2=0p_1 = p_2 = 0, and p3=1/2p_3 = 1/2. (b) p0=p1=p2=p3=1/4p_0 = p_1 = p_2 = p_3 = 1/4. (c) p0=tp_0 = t, p1=1−2tp_1 = 1 - 2t, p2=0p_2 = 0, and p3=tp_3 = t, where t≤1/2t \leq 1/2.

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