Problem 1

(a) Let U1, U2, . . . , Un be independent random variables, each with the uniform distribution on the unit interval [0, 1]. Let U(k) be the k-th smallest number in U1, . . . , Un (that is, U(k) is the k-th number in the order statistics). Find the density of U(k)

(b) Let X1, X2, . . . , Xn be independent random variables, each with exponential distribution with rate 1. Let X(k) be the k-th smallest number in X1, . . . , Xn (that is, X(k) is the k-th number in the order statistics). Find the density of X(k) .