Let X be a continuous random variable, with CDF F(x), taking values in an interval [0, b]; that is, F(0) = 0 and
F(b) = 1. Then there is an alternative formula for expected value:
E(X) = Z b0 (1 - F(x)) dx. (1)
(a) Assume b is a finite number. Prove (1) using integration-by-parts. [Hint: Recall that the PDF is f(x) = d
dxF(x).]
(b) Check that the formula (1) holds when X Unif(0, b).
(c) Formula (1) also works for b = 1. Check this when X is an Exponential RV with PDF f(x) = e-x for x 0.