1. Consider the simple birth and death process with birth and death rates defined by = , ; = for i = 0, 1, 2,... Suppose X(0) = n. (a) Use the forward Kolmogorov equations for the general birth-death model to write down differential equations for pk (t) = Pr(X(t) = k | X(0) = n), k = 0, 1, 2, . . .. (b) Show that the probability generating function G(s) = P(t)s satisfies a t Gt(s) = As. J Gt (8) - ( + ) s Gt (s) + -Gt(s) (c) By considering 2G/Ost show that d [X(t)] = (A)E [X(t)] and deduce that E[X(t)] = ne(-)t Hint: From properties of generating functions, E[X(t)] = G(1). (d) When > prove that the probability of extinction is 1. State clearly any results from the notes which you use in your proof.