[10 marks] Let {Xm n 2 0} be a branching process, with X0 = 1. The offspring distribution is given by IP(Y=0)=1p, P(Y=3)=p, for somep 6 (0,1). a) [2 marks] Find the probability generating function of Y. b) [2 marks] Find the probability that the process is extinct by the second gen- eration; that is, IP(X2 = 0). c) [4 marks] Find the probability of eventual extinction are as a function of p E (0, 1). d) [2 marks] Assume p > 1/3. Find the conditional probability of eventual extinction, given that X17 = k (k E N)