Question 4. [14 Marks] Consider a branching Markov chain (XQRED where KG 2 l and the offspring distribution is given by 1 l 1 p03} : 1(1): 519(2) : 1 The number of offspring produced by each individual is independent of the number of offspring produced by other individuals. [3) Calculate the probability of ultimate extinction. (b) Calculate the expected size (i.e. the mean size) of the nth generation. (c) Calculate the variance of the nth generation. For the remainder of the question assume that the offspring distribution is unchanged but that X0 has a Poisson distribution with PMF Am IP'[X{~j : m] : e\