In a branching process, the probability that any individual has j descendants, P; = P[Y = j] is given by Po = 0 and P; = 21" j > 1. (a) Demonstrate that the probability generating function of the first generation is G(s) = (3 marks) (b) Formulate the probability generating functions, G2(s) and G3(s), of the second and third generations respectively. (6 marks) (C) Hence, interpret and deduce an expression for the probability generating function, Gn(s), of the nth generation. (2 marks) (d) Compute P3,3 , the probability that the population size of the third generation is 3, given that the process starts with one individual. (5 marks) (e) Solve for the mean population size of the nth generation. (4 marks) In a branching process, the probability that any individual has j descendants, P; = P[Y = j] is given by Po = 0 and P; = 21" j > 1. (a) Demonstrate that the probability generating function of the first generation is G(s) = (3 marks) (b) Formulate the probability generating functions, G2(s) and G3(s), of the second and third generations respectively. (6 marks) (C) Hence, interpret and deduce an expression for the probability generating function, Gn(s), of the nth generation. (2 marks) (d) Compute P3,3 , the probability that the population size of the third generation is 3, given that the process starts with one individual. (5 marks) (e) Solve for the mean population size of the nth generation. (4 marks)