Grinstead-Snell-Probability · Problem 10.2.7

Generating Functions

Mathematics · branching-processes

Problem

Let

N

be the expected total number of offspring in a branching process. Let

m

be the mean number of offspring of a single parent. Show that

N = 1 + \left(\sum p_k \cdot k\right) N = 1 + mN

and hence that

N

is finite if and only if

m < 1

and in that case

N = 1/(1-m)