The random variable X ~ Poisson(1.1) is the offspring variate in a Galton-Watson branching process starting with a single ancestor in generation zero. (a) Find the mean number of offspring produced by an individual in the population. Decide whether eventual extinction of the process is certain. Briefly justify your decision. (b) Find the probability (correct to three decimal places) of extinction (i) by the third generation, and (ii) at the third generation. (c) Use the Newton-Raphson method to show that the iteration formula, to determine the probability of eventual extinction of this process, is Xn+1 = Xn e-1.1(1-xn) Xn 1.1e-1.1(1-n) - 1