Example 3.2

In a branching process the number of off springs per individual has a binomial distribution with parameters  and . Starting with a single individual calculate (i) the extinction probability, (ii) the probability that the population becomes extinct for the first time in the third generation. Suppose that instead of starting with a single individual the initial population size,  is a random variable that is Poisson distributed with mean, . One can easily show that in this case the extinction probability is given for   by .

Example 3.3

Consider a branching process in which the number of offspring per individual has a Poisson distribution with mean, . Let  denote the probability that, starting with a single individual the population eventually becomes extinct. Also let  ,  be such that . We can prove that (i) , (ii) conditional on eventual extinction, the branching process follows the same probability law as the branching process in which the number of offspring per individual is Poisson with mean .