Raw 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 raw moment $\expect {X^n}$ of $X$ is given by:


 * $\expect {X^n} = n! \beta^n$

Proof
From Moment Generating Function of Exponential Distribution, the moment generating function of $X$, $M_X$, is given by:


 * $\displaystyle \map {M_X} t = \frac 1 {1 - \beta t}$

By Moment in terms of Moment Generating Function:


 * $\displaystyle \expect {X^n} = \map {M^{\paren n}_X} 0$

We have:

Setting $t = 0$ gives: