Editing Beta distribution (section)

===Probability density function===
[[File:PDF of the Beta distribution.gif|thumb|An animation of the beta distribution for different values of its parameters.]]
The [[probability density function]] (PDF) of the beta distribution, for <math> 0 \leq x \leq 1 </math> or <math> 0 < x < 1 </math>, and shape parameters <math> \alpha </math>, <math> \beta > 0 </math>, is a [[power function]] of the variable <math> x </math> and of its [[Reflection formula|reflection]] <math> (1-x) </math> as follows:

: <math>
\begin{align}
f(x;\alpha,\beta) & = \mathrm{constant}\cdot x^{\alpha-1}(1-x)^{\beta-1} \\[3pt]
& = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{\displaystyle \int_0^1 u^{\alpha-1} (1-u)^{\beta-1}\, du} \\[6pt]
& = \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1} \\[6pt]
& = \frac{1}{\Beta(\alpha,\beta)} x^{\alpha-1}(1-x)^{\beta-1}
\end{align}
</math>

where <math> \Gamma(z) </math> is the [[gamma function]].  The [[beta function]], <math>\Beta</math>, is a [[normalization constant]] to ensure that the total probability is&nbsp;1. In the above equations <math> x </math> is a [[Realization (probability)|realization]]&mdash;an observed value that actually occurred&mdash;of a [[random variable]] <math> X </math>.

Several authors, including [[Norman Lloyd Johnson|N. L. Johnson]] and [[Samuel Kotz|S. Kotz]],<ref name=JKB /> use the symbols <math> p </math> and <math> q </math> (instead of <math> \alpha </math> and <math> \beta </math>) for the shape parameters of the beta distribution, reminiscent of the symbols traditionally used for the parameters of the [[Bernoulli distribution]], because the beta distribution approaches the Bernoulli distribution in the limit when both shape parameters <math> \alpha </math> and <math> \beta </math> approach zero.

In the following, a random variable <math> X </math> beta-distributed with parameters <math> \alpha </math> and <math> \beta </math> will be denoted by:<ref name="Mathematical Statistics with MATHEMATICA"/><ref name=Kruschke2011 />

:<math>X \sim \operatorname{Beta}(\alpha, \beta)</math>

Other notations for beta-distributed random variables used in the statistical literature are <math>X \sim \mathcal{B}e(\alpha, \beta)</math><ref name=BergerDecisionTheory>{{cite book |last=Berger |first=James O. |title=Statistical Decision Theory and Bayesian Analysis |edition=2nd |year=2010 |publisher=Springer |isbn=978-1441930743}}</ref> and <math>X \sim \beta_{\alpha, \beta}</math>.<ref name=Feller>{{cite book|last=Feller|first=William|title=An Introduction to Probability Theory and Its Applications, Vol. 2|year=1971|publisher=Wiley|isbn=978-0471257097|url=https://archive.org/details/introductiontopr00fell}}</ref>