n=0 5. Let (X₂)‰。 be a branching process with common distribution § having mean µ > 1. Assume that Xo = 1. Let og(s) = E[s] be the generating function of . Recall that §. the extinction probability u∞ satisfies u∞ < 1 in this case. (a) Explain why o'¿(u…) < 1. (b) Let un = P(X₂ = 0). Use part (a) to show that there exists p < 1 such that for all sufficiently large n, U∞ - Un+1 ≤ p(u∞ un). Hint: what is the formal definition of the derivative? (c) Show that there exist b > 0, c < ∞ such that for all n, P( extinction |Xn ‡ 0) ≤ ce¯bn. (As an exercise that you do not need to turn in, think of how you might interpret this inequality practically).

Fig: 1