Grinstead-Snell-Probability · Problem 10.2.1

Generating Functions

Mathematics · 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 eventually dies out if (a) p0=1/2p_0 = 1/2, p1=1/4p_1 = 1/4, and p2=1/4p_2 = 1/4. (b) p0=1/3p_0 = 1/3, p1=1/3p_1 = 1/3, and p2=1/3p_2 = 1/3. (c) p0=1/3p_0 = 1/3, p1=0p_1 = 0, and p2=2/3p_2 = 2/3. (d) pj=1/2j+1p_j = 1/2^{j+1}, for j=0,1,2,…j = 0, 1, 2, \ldots. (e) pj=(1/3)(2/3)jp_j = (1/3)(2/3)^j, for j=0,1,2,…j = 0, 1, 2, \ldots. (f) pj=e−22j/j!p_j = e^{-2}2^j/j!, for j=0,1,2,…j = 0, 1, 2, \ldots (estimate dd numerically).

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