# Raw Moment of Beta Distribution

## Theorem

Let $X \sim \BetaDist \alpha \beta$ for some $\alpha, \beta > 0$, where $\operatorname {Beta}$ is the Beta distribution.

Then:

$\ds \expect {X^n} = \prod_{r \mathop = 0}^{n - 1} \frac {\alpha + r} {\alpha + \beta + r}$

for positive integer $n$.

## Proof

By Moment Generating Function of Beta Distribution, the moment generating function, $M_X$, of $X$ is given by:

$\ds \map {M_X} t = \expect {e^{t X} } = 1 + \sum_{n \mathop = 1}^\infty \paren {\prod_{r \mathop = 0}^{n - 1} \frac {\alpha + r} {\alpha + \beta + r} } \frac {t^n} {n!}$

We also have:

 $\ds \expect {e^{t X} }$ $=$ $\ds \expect {\sum_{n \mathop = 0}^\infty \frac {t^n X^n} {n!} }$ Power Series Expansion for Exponential Function $\ds$ $=$ $\ds \sum_{n \mathop = 0}^\infty \expect {\frac {t^n X^n} {n!} }$ Linearity of Expectation Function $\ds$ $=$ $\ds \sum_{n \mathop = 0}^\infty \expect {X^n} \frac {t^n} {n!}$ Linearity of Expectation Function $\ds$ $=$ $\ds \frac {t^0} {0!} \expect {X^0} + \sum_{n \mathop = 1}^\infty \expect {X^n} \frac {t^n} {n!}$ $\ds$ $=$ $\ds 1 + \sum_{n \mathop = 1}^\infty \expect {X^n} \frac {t^n} {n!}$ Expectation of Constant

Comparing coefficients yields the result.

$\blacksquare$