* Suppose that each individual of the geobacter randomos bacterial species successfully undergoes ternary fission with probability p 0 (0, 1) and otherwise dies. Assuming that each individual behaves independently and identically, we shall model the number of individuals as a branching process (Sn, n=0, 1, ...). (a) Write down the pgf, G(z), of the offspring distribution. [1] (b) Assuming S0 = 1, determine explicit expressions for the mean and variance of Sn as a function of n. In terms of p, when is u 1 (supercritical)? [2] (c) Assuming S0 = 1, determine the probability of ultimate extinction, n, as a function of p. [1] (d) # Assuming S0 = 1, determine an explicit expression for Gn(z) = Ez Sn for n {1, 2, 3}. [1] 1 (e) # Assuming S0 = 1, determine n = P(Sn = 0) for n {1, 2, 3}. * Suppose that each individual of the geobacter randomos bacterial species successfully undergoes ternary fission with probability p 0 (0, 1) and otherwise dies. Assuming that each individual behaves independently and identically, we shall model the number of individuals as a branching process (Sn, n=0, 1, ...). (a) Write down the pgf, G(z), of the offspring distribution. [1] (b) Assuming S0 = 1, determine explicit expressions for the mean and variance of Sn as a function of n. In terms of p, when is u 1 (supercritical)? [2] (c) Assuming S0 = 1, determine the probability of ultimate extinction, n, as a function of p. [1] (d) # Assuming S0 = 1, determine an explicit expression for Gn(z) = Ez Sn for n {1, 2, 3}. [1] 1 (e) # Assuming S0 = 1, determine n = P(Sn = 0) for n {1, 2, 3}