Problem
Let $X$ be a continuous random variable with CDF $F$. Suppose that $P(X > 0) = 1$ and that $E(X)$ exists. Show that $E(X) = \int_0^{\infty} P(X > x)dx$.
Hint: Consider integrating by parts. The following fact is helpful: if $E(X)$ exists, then $lim_{x \to \infty} x[1 - F(x)] = 0$.
Solution
Sources
- 36-705 hw1 problem 10