Grinstead-Snell-Probability · Problem 6.3.13

Sums of Random Variables

Mathematics · expected-value-variance-continuous

Problem

Let

X

be a random variable that takes on nonnegative values and has distribution function

F(x)

. Show that

E(X) = \int_0^{\infty} (1 - F(x))\,dx\,.

Hint: Integrate by parts. Illustrate this result by calculating

E(X)

by this method if

X

has an exponential distribution

F(x) = 1 - e^{-\lambda x}

for

x \geq 0

, and

F(x) = 0

otherwise.