Excess Kurtosis of Beta Distribution

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

Then the excess kurtosis $\gamma_2$ of $X$ is given by:


 * $\gamma_2 = \dfrac {6 \paren {\paren {\alpha - \beta}^2 \paren {\alpha + \beta + 1} - \alpha \beta \paren {\alpha + \beta + 2} } } {\alpha \beta \paren {\alpha + \beta + 2} \paren {\alpha + \beta + 3} }$