Answer Happy

Consider a branching Markov chain (Xn)n>o where Xo = 1 and the offspring distribution is given by 1 1 p(0) = 6, P(1) = 5, p(2) = 3 The number of offspring produced by each individual is independent of the number of offspring produced by other individuals. (a) Calculate the probability of ultimate extinction. (b) Calculate the expected size (i.e. the mean size) of the n-th generation. (c) Calculate the variance of the n-th generation. For the remainder of the question assume that the offspring distribution is unchanged but that X, has a geometric distribution with PMF P[Xo = m) = (1 – 7)-1, m = 1, 2, ..., where 0 << 1. (d) Find the mean and variance of X, . (e) Calculate the probability of ultimate extinction. (f) Calculate the expected size (i.e. the mean size) of the n-th generation. (g) Calculate the variance of the n-th generation.