Let Xn be the branching process has generating function o(s) and initial population Xo = 1. Let Zn be the cumulative population up to n-th generation, i.e. n Zn = >Xk K=0 And let Un = E(s ") be the generating function for Zn. (a) Denote u = o'(1). Calculate E(Zn) for some fixed integer n. For full credit, your answer should not have summation symbol. (b) Use First step analysis, condition on all possible values of X1, show the following is true. an(s) = sp(Wn-1(s)) Show all of your work and explain important steps (c) If we know o(s) =+ Use the result in (b) to find w(s) with 1/oo (s) = lim Un(s)