# Tagged: integral by parts

## Problem 757

Let $X$ be an exponential random variable with parameter $\lambda$.

(a) For any positive integer $n$, prove that
$E[X^n] = \frac{n}{\lambda} E[X^{n-1}].$

(b) Find the expected value of $X$.

(c) Find the variance of $X$.

(d) Find the standard deviation of $X$.