Introduction to Probability · Problem 10.2.8

Generating Functions — Grinstead & Snell

Mathematics — Probability · 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

Intuitive explanation

A branching process tracks how a population evolves across generations, where each individual produces offspring according to a fixed probability distribution. The key insight is that the offspring distribution here has a special structure—it decays geometrically—which lets you use generating functions to find the distribution of population sizes at each generation. Once you have the distribution $p_j^{(n)}$ , extinction probabilities follow by tracking when the population first hits zero, and the expected lifetime emerges from summing survival probabilities across all generations.

Intuitive explanation

Related theory

branching-processes

Browse the problem library