Let [math]X[/math] be a random variable that takes on nonnegative values and has distribution function [math]F(x)[/math]. Show that

[[math]] E(X) = \int_0^\infty (1 - F(x))\, dx\ . [[/math]]

Hint: Integrate by parts. Illustrate this result by calculating [math]E(X)[/math] by this method if [math]X[/math] has an exponential distribution [math]F(x) = 1 - e^{-\lambda x}[/math] for [math]x \geq 0[/math], and [math]F(x) = 0[/math] otherwise.