Grinstead-Snell-Probability · Problem 10.2.8

Generating Functions

Mathematics · branching-processes

Problem

Consider a branching process such that the number of offspring of a parent is

j

with probability

1/2^{j+1}

for

j = 0, 1, 2, \ldots

. (a) Using the results of Example 10.11 show that the probability that there are

j

offspring in the

n

th generation is

p_j^{(n)} = \begin{cases} \frac{1}{n(n+1)}\left(\frac{n}{n+1}\right)^j, & \text{if } j \geq 1, \\ \frac{n}{n+1}, & \text{if } j = 0. \end{cases}

(b) Show that the probability that the process dies out exactly at the

n

th generation is

1/n(n+1)

. (c) Show that the expected lifetime is infinite even though

d = 1