Consider a population consisting of individuals able to produce offspring of the same kind...

Consider a population consisting of individuals able to produce offspring of the same kind. Suppose that by the end of its lifetime, each individual will have produced j new offspring with probability Pj, j ≥ 0, independently of the number produced by any other individual. The number of individuals initially present, denoted by X0, is called the size of the zeroth generation. All offspring of the zeroth generation constitute the first generation, and their number is denoted by X1. In general, let Xn denote the size of the nth generation. Let and denote, respectively, the mean and the variance of the number of offspring produced by a single individual. Suppose that X0 = 1- that is, initially there is a single individual in the population.

(a) Show that

---

(b) Use part (a) to conclude that

---

(c) Show that

---

(d) Use part (c) to conclude that

---

The model just described is known as a branching process, and an important question for a population that evolves along such lines is the probability that the population will eventually die out. Let π denote this probability when the population starts with a single individual. That is,π = P{population eventually dies out | X0 = 1)

---

(e) Argue that π satisfies