Given (X_{0}=1), show that the probability of extinction (pi_{0}) satisfies equation [Mleft(pi_{0} ight)=pi_{0}] by applying the total

Given \(X_{0}=1\), show that the probability of extinction \(\pi_{0}\) satisfies equation

\[M\left(\pi_{0}\right)=\pi_{0}\]

by applying the total probability rule (condition with regard to the number of offspring of the individual in the zerouth generation). Make use of the answer to exercise 8.28.

Data from answer of Exercise 8.28

Transcribed Image Text:

The offspring of the n individuals generates n independent Galton-Watson processes. Each of them has extinction probability o. Therefore, the extinction probability of a Galton-Watson process starting with n individuals is .