16 Let X be a nonnegative random variable with distribution function F. Define 1(t)= = { b 1 if X > t 10 otherwise.

(a) Prove that I(t)dt = X.

(b) By calculating the expected value of both sides of part (a), prove that E(X) = (1-F(1)]dt. This is a special case of Theorem 6.2.

(c) For r> 0, use part

(b) to prove that E(X) = r-[1 F(r)]dt. -