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4) Consider a branching process {X1l . . . ,X}. Let Pi = P(X1 = j |X0 = 1), and we know that R; = 0.5.131 = 0.3, and P2 = 0.2. (a) What is ]E(X1 | X0 = 1) and Va'r(X1 | X0 = 1)? (b) What is ]E(X1 | X]; = 10,000) and Var-(X1 | X0 = 10,000) (Hint: X1 = 211:)?" Zk, where Zk is defined in the text book or the lecture note)? (c) Suppose that X0 = 1. What is the probability that the population dies out? (d) Suppose that the population of a small country is 10, 000. The birth probability is the same with above. What is the probability that the population of this country dies out? (e) Suppose that this country begins to give a strong incentive to increase the birth rate, and now the birth probability is Pb : 0.2,P1 : 0.5, P2 : 0.3. Then, what is the probability that the population of this country dies out