Moment Generating Function of Beta Distribution

Theorem

Let $X \sim \BetaDist \alpha \beta$ denote the Beta distribution fior some $\alpha, \beta > 0$.

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

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

Proof

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

$\map {f_X} x = \dfrac {x^{\alpha - 1} \paren {1 - x}^{\beta - 1} } {\map \Beta {\alpha, \beta} }$

From the definition of a moment generating function:

$\ds \map {M_X} t = \expect {e^{t X} } = \int_0^1 e^{t x} \map {f_X} x \rd x$

So:

\(\ds \map {M_X} t\)

\(=\)

\(\ds \frac 1 {\map \Beta {\alpha, \beta} } \int_0^1 e^{t x} x^{\alpha - 1} \paren {1 - x}^{\beta - 1} \rd x\)

\(\ds \)

\(=\)

\(\ds \frac 1 {\map \Beta {\alpha, \beta} } \int_0^1 \paren {\sum_{k \mathop = 0}^\infty \frac {\paren {t x}^k} {k!} } x^{\alpha - 1} \paren {1 - x}^{\beta - 1} \rd x\)

Power Series Expansion for Exponential Function

\(\ds \)

\(=\)

\(\ds \frac 1 {\map \Beta {\alpha, \beta} } \sum_{k \mathop = 0}^\infty \frac {t^k} {k!} \int_0^1 x^{\alpha + k - 1} \paren {1 - x}^{\beta - 1} \rd x\)

Power Series is Termwise Integrable within Radius of Convergence

\(\ds \)

\(=\)

\(\ds \sum_{k \mathop = 0}^\infty \frac{t^k} {k!} \paren {\frac {\map \Beta {\alpha + k, \beta} } {\map \Beta {\alpha, \beta} } }\)

Definition of Beta Function

\(\ds \)

\(=\)

\(\ds \frac {\map \Beta {\alpha, \beta} } {\map \Beta {\alpha, \beta} } \frac {t^0} {0!} + \sum_{k \mathop = 1}^\infty \frac{t^k} {k!} \paren {\frac {\map \Beta {\alpha + k, \beta} } {\map \Beta {\alpha, \beta} } }\)

\(\ds \)

\(=\)

\(\ds 1 + \sum_{k \mathop = 1}^\infty \paren {\frac {\map \Gamma {\alpha + k} \map \Gamma \beta} {\map \Gamma {\alpha + \beta + k} } \cdot \frac {\map \Gamma {\alpha + \beta} } {\map \Gamma \alpha \map \Gamma \beta} } \frac{t^k} {k!}\)

Definition 3 of Beta Function

\(\ds \)

\(=\)

\(\ds 1 + \sum_{k \mathop = 1}^\infty \paren {\frac {\map \Gamma {\alpha + k} } {\map \Gamma \alpha} \cdot \frac {\map \Gamma {\alpha + \beta} } {\map \Gamma {\alpha + \beta + k} } } \frac{t^k} {k!}\)

\(\ds \)

\(=\)

\(\ds 1 + \sum_{k \mathop = 1}^\infty \paren {\frac {\map \Gamma \alpha \prod_{r \mathop = 0}^k \paren {\alpha + r} } {\map \Gamma \alpha} \cdot \frac {\map \Gamma {\alpha + \beta} } {\map \Gamma {\alpha + \beta} \prod_{r \mathop = 0}^k \paren {\alpha + \beta + r} } } \frac{t^k} {k!}\)

Gamma Difference Equation

\(\ds \)

\(=\)

\(\ds 1 + \sum_{k \mathop = 1}^\infty \paren {\prod_{r \mathop = 0}^{k - 1} \frac {\alpha + r} {\alpha + \beta + r} } \frac{t^k} {k!}\)

Product of Products

$\blacksquare$