Editing Beta distribution (section)

===Skewness===
[[File:Skewness for Beta Distribution as a function of the variance and the mean - J. Rodal.jpg|325px|thumb|Skewness for beta distribution as a function of variance and mean]]

The [[skewness]] (the third moment centered on the mean, normalized by the 3/2 power of the variance) of the beta distribution is<ref name=JKB />

:<math>\gamma_1 =\frac{\operatorname{E}[(X - \mu)^3]}{(\operatorname{var}(X))^{3/2}} = \frac{2(\beta - \alpha)\sqrt{\alpha + \beta + 1}}{(\alpha + \beta + 2) \sqrt{\alpha \beta}} .</math>

Letting ''α'' = ''β'' in the above expression one obtains ''γ''<sub>1</sub> = 0, showing once again that for ''α'' = ''β'' the distribution is symmetric and hence the skewness is zero. Positive skew (right-tailed) for ''α'' < ''β'', negative skew (left-tailed) for ''α'' > ''β''.

Using the [[Statistical parameter|parametrization]] in terms of mean ''μ'' and sample size ''ν'' = ''α'' + ''β'':

:<math> \begin{align}
  \alpha & {} = \mu \nu ,\text{ where }\nu =(\alpha + \beta)  >0\\
  \beta & {} = (1 - \mu) \nu , \text{ where }\nu =(\alpha + \beta)  >0.
\end{align}</math>

one can express the skewness in terms of the mean ''μ'' and the sample size ν as follows:

:<math>\gamma_1 =\frac{\operatorname{E}[(X - \mu)^3]}{(\operatorname{var}(X))^{3/2}} = \frac{2(1-2\mu)\sqrt{1+\nu}}{(2+\nu)\sqrt{\mu (1 - \mu)}}.</math>

The skewness can also be expressed just in terms of the variance ''var'' and the mean ''μ'' as follows:

:<math>\gamma_1 =\frac{\operatorname{E}[(X - \mu)^3]}{(\operatorname{var}(X))^{3/2}} = \frac{2(1-2\mu)\sqrt{\operatorname{var}}}{ \mu(1-\mu) + \operatorname{var}}\text{ if } \operatorname{var} < \mu(1-\mu)</math>

The accompanying plot of skewness as a function of variance and mean shows that maximum variance (1/4) is coupled with zero skewness and the symmetry condition (''μ'' = 1/2), and that maximum skewness (positive or negative infinity) occurs when the mean is located at one end or the other, so that the "mass" of the probability distribution is concentrated at the ends (minimum variance).

The following expression for the square of the skewness, in terms of the sample size ''ν'' = ''α'' + ''β'' and the variance var, is useful for the method of moments estimation of four parameters:

:<math>(\gamma_1)^2 =\frac{(\operatorname{E}[(X - \mu)^3])^2}{(\operatorname{var}(X))^3} = \frac{4}{(2+\nu)^2}\bigg(\frac{1}{\operatorname{var}}-4(1+\nu)\bigg)</math>

This expression correctly gives a skewness of zero for ''α'' = ''β'', since in that case (see {{section link||Variance}}): <math>\operatorname{var} = \frac{1}{4 (1 + \nu)}</math>.

For the symmetric case (''α'' = ''β''), skewness = 0 over the whole range, and the following limits apply:

:<math>\lim_{\alpha = \beta \to 0} \gamma_1 = \lim_{\alpha = \beta \to \infty} \gamma_1 =\lim_{\nu \to 0} \gamma_1=\lim_{\nu \to \infty} \gamma_1=\lim_{\mu \to \frac{1}{2}} \gamma_1 = 0</math>

For the asymmetric cases (''α'' ≠ ''β'') the following limits (with only the noted variable approaching the limit) can be obtained from the above expressions:

:<math> \begin{align}
&\lim_{\alpha\to  0} \gamma_1 =\lim_{\mu\to  0} \gamma_1 = \infty\\
&\lim_{\beta \to  0} \gamma_1  = \lim_{\mu\to  1} \gamma_1= - \infty\\
&\lim_{\alpha\to \infty} \gamma_1 = -\frac{2}{\sqrt\beta},\quad \lim_{\beta \to  0}(\lim_{\alpha\to  \infty} \gamma_1) = -\infty,\quad \lim_{\beta \to \infty}(\lim_{\alpha\to \infty} \gamma_1) = 0\\
&\lim_{\beta\to  \infty} \gamma_1 = \frac{2}{\sqrt\alpha},\quad \lim_{\alpha \to  0}(\lim_{\beta \to  \infty} \gamma_1) = \infty,\quad \lim_{\alpha \to  \infty}(\lim_{\beta \to  \infty} \gamma_1) = 0\\
&\lim_{\nu \to  0} \gamma_1 = \frac{1 - 2 \mu}{\sqrt{\mu (1-\mu)}},\quad \lim_{\mu \to  0}(\lim_{\nu \to  0} \gamma_1)  = \infty,\quad \lim_{\mu \to  1}(\lim_{\nu \to  0} \gamma_1) = - \infty
\end{align}</math>

[[File:Skewness Beta Distribution for alpha and beta from 1 to 5 - J. Rodal.jpg|325px]]
[[File:Skewness Beta Distribution for alpha and beta from .1 to 5 - J. Rodal.jpg|325px]]