Consider the branching process of Example XXXXXXXXXXIn that process, before death an organism produces j (j ≥ 0) offspring with probability α j . Let X 0 = 1, and let µ, the expected number of...

Consider the branching process of Example 12.24. In that process, before death an organism produces j (j ≥ 0) offspring with probability α_j. Let X₀
= 1, and let µ, the expected number of offspring of an organism, be greater than 1. Let p be the probability that extinction will occur. Show that p is the smallest positive root of the equation

Example 12.24

Suppose that before death an organism produces j offspring with probability α_j, (j ≥ 0) independently of other organisms. Let X₀
be the size of the initial population of such organisms. The number of all offspring of the initial population, denoted by X₁, is the population size at the first generation. All offspring of the first generation form the second generation, and the population