Let X be a discrete random variable with probability function fx (x), and suppose that a X I). I=a (a) Show that (1- 2)Tx(2) = 2"-Gx(2), where Gx(2) is the probability generating function of X. In particular, if X is a non-negative discrete random variable, show that (1 - z)Tx(2) =1 - Gx(2). (b) Using the result from (a) for a non-negative discrete random variable X, show that E(X) = Tx (1) and var(X) = 2TY(1) + Tx (1) -Tx(1)2. (c) Let random variables (Yi, i 2 1) be independently and uniformly distributed on {1, 2, . . ., n}. Let Sk = > Ya, and define In = min(k : Sk > n}. Thus Tn is the smallest number of the Y, required to achieve a sum exceeding n. Show that P(S; }_ 2 P(Tn > j). (f) Using the results from (b) and (e), calculate E(7,) and var(Tn). (g) Find the probability generating function GT, (2). (h) Find the probability function of Tn. (i) What is the limiting probability function of T as n - co