8. (Total: 11 points)
(a) (5 points) Suppose that the random variable X takes on values of 0, 1, 2,..., and E(X) exists. Show that
$\infty$
E(X) = \sum_{n=0} Pr(X > n).
(b) (6 points) There are three different types of coupons. Each time a person collects a coupon and it is, independent of those previously obtained, a type j coupon with probability $p_j$ where j = 1, 2, 3, and $p_1 + p_2 + p_3 = 1$. Let N denote the number of coupons that one needs to collect in order to have a complete collection of at least one of each type. Find E(N).
