Central Moment of Exponential Distribution

Theorem
Let $X$ be a continuous random variable of the exponential distribution with parameter $\beta$ for some $\beta \in \R_{> 0}$

Let $n$ be a strictly positive integer.

Then the $n$th central moment $\mu_n$ of $X$ is given by:


 * $\displaystyle \mu_n = n! \beta^n \sum_{k \mathop = 0}^n \frac {\paren {-1}^k} {k!}$

Proof
From definition of central moment we have:
 * $\mu_n = \expect {\paren {x - \mu}^n}$

By Expectation of Exponential Distribution we have:
 * $\mu = \beta$

So: