a) If X1...Xn are independent U(0,1) random variables and Y = ( ∏ from i=1 to n of Xi)^1/n, show that nlogY has m.g.f. (1 + z)^-n. i=1 (Hint: first find the m.g.f. of log Xi.
(b) Suppose that Y has density [n^n/(n-1)!](-y log y)^n-1 on (0,1). Show that E(Y^nz) =(1 + z)^-n. (Hint: start by substituting t = -log y.) (Here you may use the fact that integral form 0 to infinity of (s^n-1)ds = (n-1)! for n = 1, 2...)
(c) Deduce that the random variable Y in (a) has the density given in (b).