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:


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

Proof
From Kurtosis in terms of Non-Central Moments, we have:


 * $\gamma_2 = \dfrac {\expect {X^4} - 4 \mu \expect {X^3} + 6 \mu^2 \expect {X^2} - 3 \mu^4} {\sigma^4} - 3$

where:
 * $\mu$ is the expectation of $X$.
 * $\sigma$ is the standard deviation of $X$.

We have, by Expectation of Beta Distribution:


 * $\expect X = \dfrac {\alpha} {\alpha + \beta}$

By Variance of Beta Distribution:


 * $\var X = \sigma^2 = \dfrac {\alpha \beta} {\paren {\alpha + \beta}^2 \paren {\alpha + \beta + 1} }$

so:


 * $\sigma = \dfrac {\sqrt {\alpha \beta} } {\paren {\alpha + \beta} \paren {\sqrt {\alpha + \beta + 1 } } }$

From Raw Moment of Beta Distribution, we have:

Hence:

Relying upon Lemma 1, Lemma 2, Lemma 3 and Lemma 4 to simplify the $80$ terms in the numerator, we obtain:

Putting our simplified numerator back in, we obtain: