Skewness of Beta Distribution

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

Then the skewness $\gamma_1$ of $X$ is given by:


 * $\gamma_1 = \dfrac {2 \paren {\beta - \alpha} \sqrt {\alpha + \beta + 1} } {\paren {\alpha + \beta + 2} \sqrt {\alpha \beta} }$

Proof
From Skewness in terms of Non-Central Moments:


 * $\gamma_1 = \dfrac {\expect {X^3} - 3 \mu \sigma^2 - \mu^3} {\sigma^3}$

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:

So..