Question 5 Let X be a discrete random variable with probability function fx (x), and suppose that as X Sb. Define the tail generating function Tx() = P(X >). Show that (1-2)Tx(+) = "-Gx(), where Gx(+) is the probability generating function of x. In particular, if X is a non-negative discrete random variable, show that (1 - 2)Tx(x) = 1 - Gx(x). brusing the result from (a) for a non-negative discrete random variable X, show that E(X) = Tx(1) and var(X) = 2TX (1) + Tx(1) - Tx (1) N Let random variables (Y,i > 1) be independently and uniformly distributed on {1,2,...,n). Let Sk = Ek Y., and define Tn = min{k : S% > n}. Thus Tn is the smallest number of the Y; required to achieve a sum exceeding n. Show that P(S, Sn) () קז Show that in j+1 if and only if S, Sn. Tej Find the tail generating function T.. () ) P(In> ). Using the results from (b) and (e), calculate E(T) and var(Tn). (g) Find the probability generating function Gr. (2). (h) Find the probability function of Tr. (1) What is the limiting probability function of Tn as n + ?