# Central Moment of Exponential Distribution

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## 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:

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