Raw Moment of Beta Distribution

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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.

$\blacksquare$