Editing Beta distribution (section)

====Variance====
The [[variance]] (the second moment centered on the mean) of a beta distribution [[random variable]] ''X'' with parameters ''α'' and ''β'' is:<ref name=JKB /><ref>{{cite web | url = http://www.itl.nist.gov/div898/handbook/eda/section3/eda366h.htm | title = NIST/SEMATECH e-Handbook of Statistical Methods 1.3.6.6.17. Beta Distribution | website = [[National Institute of Standards and Technology]] Information Technology Laboratory | access-date = May 31, 2016 |date = April 2012 }}</ref>

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

Letting ''α'' = ''β'' in the above expression one obtains

:<math>\operatorname{var}(X) = \frac{1}{4(2\beta + 1)},</math>

showing that for ''α'' = ''β'' the variance decreases monotonically as {{nowrap|1=''α'' = ''β''}} increases. Setting {{nowrap|1=''α'' = ''β'' = 0}} in this expression, one finds the maximum variance var(''X'') = 1/4<ref name=JKB /> which only occurs approaching the limit, at {{nowrap|1=''α'' = ''β'' = 0}}.

The beta distribution may also be [[Statistical parameter|parametrized]] in terms of its mean ''μ'' {{nowrap|1=(0 < ''μ'' < 1)}} and sample size {{nowrap|1=''ν'' = ''α'' + ''β''}} ({{nowrap|''ν'' > 0}}) (see subsection  [[#Mean and sample size|Mean and sample size]]):

:<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>

Using this [[Statistical parameter|parametrization]], one can express the variance in terms of the mean ''μ'' and the sample size ''ν'' as follows:

:<math>\operatorname{var}(X) = \frac{\mu (1-\mu)}{1 + \nu}</math>

Since {{nowrap|1=''ν'' = ''α'' + ''β'' > 0}}, it follows that {{nowrap|var(''X'') < ''μ''(1 − ''μ'')}}.

For a symmetric distribution, the mean is at the middle of the distribution, {{nowrap|1=''μ'' = 1/2 }}, and therefore:

:<math>\operatorname{var}(X) = \frac{1}{4 (1 + \nu)} \text{ if } \mu = \tfrac{1}{2}</math>

Also, the following limits (with only the noted variable approaching the limit) can be obtained from the above expressions:

:<math> \begin{align}
&\lim_{\beta\to 0} \operatorname{var}(X) =\lim_{\alpha \to 0} \operatorname{var}(X) =\lim_{\beta\to \infty} \operatorname{var}(X) =\lim_{\alpha \to  \infty} \operatorname{var}(X) = \lim_{\nu \to  \infty} \operatorname{var}(X) =\lim_{\mu \to  0} \operatorname{var}(X) =\lim_{\mu \to  1} \operatorname{var}(X) = 0\\
&\lim_{\nu \to  0} \operatorname{var}(X) = \mu (1-\mu)
\end{align}</math>

[[File:Variance for Beta Distribution for alpha and beta ranging from 0 to 5 - J. Rodal.jpg|325px]]