Question 9. Let X be a random variable having density given by f(x; , ) = ( (x+)+1 , if x > 0 0, elsewhere where > 0 and > 0 are parameters. In such a case it is denoted X f(x; , )

(a) (5 pts) If > 1, show that E [X] = 1 .

(b) (5 pts) Let k N such that > k. If X f(x; , ) and Y f(y; 1, ), where f is as above, show that E h X k i = k 1 E h Y k1 i .

(c) (5 pts) Deduce a general formula for E Xk when > k.