5. Let X1,X2, . . . ,Xn be iid nonnegative RVs of the continuous type. If E|X| <∞, show that E|X(r)

| <∞. Write Mn = X(n) =max(X1,X2, . . . ,Xn). Show that EMn = EMn−1+

∞

0 Fn−1(x)[1−F(x)]dx, n = 2,3, . . . .

Find EMn in each of the following cases:

(a) Xi have the common DF F(x) = 1−e−βx, x ≥ 0.

(b) Xi have the common DF F(x) = x,0 < x < 1.