1. The statistician Piggy has to wait an amount of time T0 at the post office on an occasion when she is in a great hurry. In order to investigate whether or not chance makes her wait particularly long when she is in a hurry, she checks how many visits she makes to the post office until she has to wait longer than the first time. Formally, let T1, T2, . . . be the successive waiting times and N be the number of times until some Tk > T0, that is, {N = k} = {Tj  ≤ T0, 1 ≤ j k, Tk  > T0}. What is the distribution of N under the assumption that {Tn, n ≥ 0} are i.i.d. continuous random variables? What can be said about E N ?

2. Let X1, X2, . . . , Xn be independent, continuous random variables with common distribution function F(x), and consider the order statistic (X(1), X(2), . . . , X(n)). Compute E(F (X(n)) - F (X(1))).