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:

\(\ds \expect {X^3}\)

\(=\)

\(\ds \prod_{r \mathop = 0}^2 \frac {\alpha + r} {\alpha + \beta + r}\)

\(\ds \)

\(=\)

\(\ds \paren {\dfrac {\alpha} {\alpha + \beta} } \paren {\dfrac {\alpha + 1} {\alpha + \beta + 1} } \paren {\dfrac {\alpha + 2} {\alpha + \beta + 2} }\)

So..

\(\ds \gamma_1\)

\(=\)

\(\ds \frac {\paren {\dfrac {\alpha} {\alpha + \beta} } \paren {\dfrac {\alpha + 1} {\alpha + \beta + 1} } \paren {\dfrac {\alpha + 2} {\alpha + \beta + 2} } - 3 \paren {\dfrac {\alpha} {\alpha + \beta} } \paren {\dfrac {\alpha \beta} {\paren {\alpha + \beta}^2 \paren {\alpha + \beta + 1} } } - \paren {\dfrac {\alpha} {\alpha + \beta} }^3} {\paren {\dfrac {\sqrt {\alpha \beta} } {\paren {\alpha + \beta} \paren {\sqrt {\alpha + \beta + 1 } } } }^3 }\)

\(\ds \)

\(=\)

\(\ds \frac {\paren {\dfrac {\alpha \paren {\alpha + 1} \paren {\alpha + 2} \paren {\alpha + \beta}^2 - 3 \alpha^2 \beta \paren {\alpha + \beta + 2} - \alpha^3 \paren {\alpha + \beta + 1} \paren {\alpha + \beta + 2} } {\paren {\alpha + \beta}^3 \paren {\alpha + \beta + 1} \paren {\alpha + \beta + 2} } } } {\paren {\dfrac {\paren {\alpha \beta} \sqrt {\alpha \beta} } {\paren {\alpha + \beta}^3 \paren {\paren {\alpha + \beta + 1} \sqrt {\alpha + \beta + 1 } } } } }\)

Expanding terms. Addition of Fractions

\(\ds \)

\(=\)

\(\ds \dfrac {\paren { \paren {\alpha + 1} \paren {\alpha + 2} \paren {\alpha + \beta}^2 - 3 \alpha \beta \paren {\alpha + \beta + 2} - \alpha^2 \paren {\alpha + \beta + 1} \paren {\alpha + \beta + 2} } \paren {\sqrt {\alpha + \beta + 1 } } } { \beta \paren {\alpha + \beta + 2} \sqrt {\alpha \beta} }\)

Canceling $\alpha$, $\paren {\alpha + \beta}^3$ and $\paren {\alpha + \beta + 1}$ from numerator and denominator

\(\ds \)

\(=\)

\(\ds \dfrac {\paren { \paren {\alpha + 1} \paren {\alpha + 2} \paren {\alpha + \beta}^2 - 3 \alpha \beta \paren {\alpha + \beta + 2} - \alpha^2 \paren {\paren {\alpha + \beta }^2 + 3 \paren {\alpha + \beta } + 2 } } \paren {\sqrt {\alpha + \beta + 1 } } } { \beta \paren {\alpha + \beta + 2} \sqrt {\alpha \beta} }\)

Grouping the $\paren {\alpha + \beta}$ term.

\(\ds \)

\(=\)

\(\ds \dfrac {\paren { \paren {\alpha + \beta}^2 \paren {\alpha^2 + 3 \alpha + 2 - \alpha^2 } - 3 \alpha \beta \paren {\alpha + \beta + 2} - \alpha^2 \paren {3 \paren {\alpha + \beta } + 2 } } \paren {\sqrt {\alpha + \beta + 1 } } } { \beta \paren {\alpha + \beta + 2} \sqrt {\alpha \beta} }\)

Grouping $\paren {\alpha + \beta}^2$

\(\ds \)

\(=\)

\(\ds \dfrac {\paren { \paren {\alpha + \beta}^2 \paren {3 \alpha + 2 } - 3 \alpha \beta \paren {\paren {\alpha + \beta } + 2} - \alpha^2 \paren {3 \paren {\alpha + \beta } + 2 } } \paren {\sqrt {\alpha + \beta + 1 } } } { \beta \paren {\alpha + \beta + 2} \sqrt {\alpha \beta} }\)

$\alpha^2$ cancels.

\(\ds \)

\(=\)

\(\ds \dfrac {\paren { \paren {\alpha + \beta} \paren {\paren {\alpha + \beta} \paren {3 \alpha + 2 } - 3 \alpha \beta - 3 \alpha^2 } - 6 \alpha \beta - 2\alpha^2 } \paren {\sqrt {\alpha + \beta + 1 } } } { \beta \paren {\alpha + \beta + 2} \sqrt {\alpha \beta} }\)

Factoring out $\paren {\alpha + \beta}$ again

\(\ds \)

\(=\)

\(\ds \dfrac {\paren { \paren {\alpha + \beta} \paren {\paren {3 \alpha^2 + 2 \alpha + 3 \alpha \beta + 2 \beta } - 3 \alpha \beta - 3 \alpha^2 } - 6 \alpha \beta - 2\alpha^2 } \paren {\sqrt {\alpha + \beta + 1 } } } { \beta \paren {\alpha + \beta + 2} \sqrt {\alpha \beta} }\)

Expanding the product

\(\ds \)

\(=\)

\(\ds \dfrac {\paren { \paren {\alpha + \beta} \paren {2 \alpha + 2 \beta } - 6 \alpha \beta - 2\alpha^2 } \paren {\sqrt {\alpha + \beta + 1 } } } { \beta \paren {\alpha + \beta + 2} \sqrt {\alpha \beta} }\)

Canceling $3 \alpha^2$ and $3 \alpha \beta$

\(\ds \)

\(=\)

\(\ds \dfrac {\paren { \paren {2 \alpha^2 + 2 \alpha \beta + 2 \alpha \beta + 2\beta^2} - 6 \alpha \beta - 2\alpha^2 } \paren {\sqrt {\alpha + \beta + 1 } } } { \beta \paren {\alpha + \beta + 2} \sqrt {\alpha \beta} }\)

Expanding the product

\(\ds \)

\(=\)

\(\ds \dfrac {\paren {2\beta^2 - 2 \alpha \beta } \paren {\sqrt {\alpha + \beta + 1 } } } { \beta \paren {\alpha + \beta + 2} \sqrt {\alpha \beta} }\)

Canceling $2 \alpha^2$ and grouping $\alpha \beta$ terms

\(\ds \)

\(=\)

\(\ds \frac {2 \paren {\beta - \alpha} \sqrt {\alpha + \beta + 1} } {\paren {\alpha + \beta + 2} \sqrt {\alpha \beta} }\)

Canceling $\beta$ from numerator and denominator

$\blacksquare$