85. Suppose the number of children born to an individual has pmf p(x). A Galton–Watson branching process unfolds as follows: At time t  0, the population consists of a single individual. Just prior to time t1, this individual gives birth to X1 individuals according to the pmf p(x), so there are X1 individuals in the rst generation. Just prior to time t  2, each of these X1 individuals gives birth independently of the others according to the pmf p(x), resulting in X2 individuals in the second generation (e.g., if X1  3, then X2  Y1  Y2  Y3, where Yi is the number of progeny of the ith individual in the rst generation).

This process then continues to yield a third generation of size X3, and so on.

a. If X1  3, Y1  4, Y2  0, Y3  1, draw a tree diagram with two generations of branches to represent this situation.

b. Let A be the event that the process ultimately becomes extinct (one way for A to occur would be to have X1  3 with none of these three second-

generation individuals having any progeny)

and let p*  P(A). Argue that p* satis es the equation That is, p*  h(p*) where h(s) is the probability generating function introduced in Exercise 138 from Chapter 3. Hint: A   (A  {X1  x}), so the law of total probability can be applied. Now given that X13, A will occur if and only if each of the three separate branching processes starting from the rst generation ultimately becomes extinct;

what is the probability of this happening?

p*  a 1p*2x # p1x2

c. Verify that one solution to the equation in

(b) is p*  1. It can be shown that this equation has just one other solution, and that the probability of ultimate extinction is in fact the smaller of the two roots. If p(0)  .3, p(1)  .5, and p(2)  .2, what is p*? Is this consistent with the value of m, the expected number of progeny from a single individual? What happens if p(0) .2, p(1) .5, and p(2)  .3?