Justify or prove all answers.

(20 points) In a branching process, an individual has 0, 1 or 2 descendants with proba bility 1/8, 3/8, 1/2, respectively. Start the process at generation 0 with one individual. (a) Compute the expected population size at generation 2. (b) Write an expression for the probability that the branching process dies out at or before generation 3. Do not evaluate. (c) Compute the probability that the branching process ever dies out. (d) Compute the probability that the process dies out conditioned on the event that the population at generation 3 consists of 4 individuals. (e) Compute the expectation of the population size at generation 4, conditioned on the event that the population size at generation 3 is equal to 4. (20 points) Consider the branching process from question 2. Let Xn be the population size at generation n = 0, 1, 2, .... Consider the associated Markov chain Xo, X1, X2, . . . with Xo = 1. (a) Write the state space of the Markov chain and the distribution of X1. (b) Determine the classes. Is this Markov chain irreducible? (c) Determine the transition probability Pij for j = 0 and any 2 2 0. (d) Compute the probability that the branching process dies out at or before gener- ation 2