Moment Generating Function of Beta Distribution

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Theorem

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

Then the moment generating function $M_X$ of $X$ is given by:

$\displaystyle M_X \left({t}\right) = 1 + \sum_{k \mathop = 1}^\infty \left({ \prod_{r \mathop = 0}^{k - 1} \frac {\alpha + r} {\alpha + \beta + r} }\right) \frac{t^k} {k!}$


Proof

From the definition of the Beta distribution, $X$ has probability density function:

$\displaystyle f_X \left({x}\right) = \frac { x^{\alpha - 1} \left({1 - x}\right)^{\beta - 1} } {\Beta \left({\alpha, \beta}\right)}$

From the definition of a moment generating function:

$\displaystyle M_X \left({t}\right) = \mathbb E \left[{ e^{t X} }\right] = \int_0^1 e^{tx} f_X \left({x}\right) \rd x$

So:

\(\ds M_X \left({t}\right)\) \(=\) \(\ds \frac 1 {\Beta \left({\alpha, \beta}\right)} \int_0^1 e^{t x} x^{\alpha - 1} \left({1 - x}\right)^{\beta - 1} \rd x\)
\(\ds \) \(=\) \(\ds \frac 1 {\Beta \left({\alpha, \beta}\right)} \int_0^1 \left({ \sum_{k \mathop = 0}^\infty \frac {\left({t x}\right)^k} {k!} }\right) x^{\alpha - 1} \left({1 - x}\right)^{\beta - 1} \rd x\) Power Series Expansion for Exponential Function
\(\ds \) \(=\) \(\ds \frac 1 {\Beta \left({\alpha, \beta}\right)} \sum_{k \mathop = 0}^\infty \frac{t^k} {k!} \int_0^1 x^{\alpha + k - 1} \left({1 - x}\right)^{\beta - 1} \rd x\) Power Series is Termwise Integrable within Radius of Convergence
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \frac{t^k} {k!} \left({ \frac {\Beta \left({\alpha + k, \beta}\right)} {\Beta \left({\alpha, \beta}\right)} }\right)\) Definition of Beta Function
\(\ds \) \(=\) \(\ds \frac{\Beta \left({\alpha, \beta}\right)} {\Beta \left({\alpha, \beta}\right)} \frac {t^0} {0!} + \sum_{k \mathop = 1}^\infty \frac{t^k} {k!} \left({ \frac {\Beta \left({\alpha + k, \beta}\right)} {\Beta \left({\alpha, \beta}\right)} }\right)\)
\(\ds \) \(=\) \(\ds 1 + \sum_{k \mathop = 1}^\infty \left({ \frac{\Gamma \left({\alpha + k}\right) \Gamma \left({\beta}\right)} {\Gamma \left({\alpha + \beta + k}\right)} \cdot \frac {\Gamma \left({\alpha+\beta}\right)} {\Gamma \left({\alpha}\right) \Gamma \left({\beta}\right)} }\right) \frac{t^k} {k!}\) Definition of Beta Function
\(\ds \) \(=\) \(\ds 1 + \sum_{k \mathop = 1}^\infty \left({ \frac{\Gamma \left({\alpha + k}\right)} {\Gamma \left({\alpha}\right)} \cdot \frac {\Gamma \left({\alpha+\beta}\right)} {\Gamma \left({\alpha + \beta + k}\right)} }\right) \frac{t^k} {k!}\)
\(\ds \) \(=\) \(\ds 1 + \sum_{k \mathop = 1}^\infty \left({\frac {\Gamma \left({\alpha}\right) \prod_{r \mathop = 0}^k \left({\alpha + r}\right)} {\Gamma \left({\alpha}\right)} \cdot \frac {\Gamma \left({\alpha + \beta}\right)} {\Gamma \left({\alpha + \beta}\right) \prod_{r \mathop = 0}^k \left({\alpha + \beta + r}\right)} }\right) \frac{t^k} {k!}\) Gamma Difference Equation
\(\ds \) \(=\) \(\ds 1 + \sum_{k \mathop = 1}^\infty \left({ \prod_{r \mathop = 0}^{k - 1} \frac {\alpha + r} {\alpha + \beta + r} }\right) \frac{t^k} {k!}\) Product of Products

$\blacksquare$