Editing Beta distribution (section)

==Random variate generation==
{{further|Non-uniform random variate generation}}

If ''X'' and ''Y'' are independent, with <math>X \sim \Gamma(\alpha, \theta)</math> and <math>Y \sim \Gamma(\beta, \theta)</math> then

:<math>\frac{X}{X+Y} \sim \Beta(\alpha, \beta).</math>

So one algorithm for generating beta variates is to generate <math>\frac{X}{X + Y}</math>, where ''X'' is a [[Gamma distribution#Random variate generation|gamma variate]] with parameters (α, 1) and ''Y'' is an independent gamma variate with parameters (β, 1).<ref>van der Waerden, B. L., "Mathematical Statistics", Springer, {{ISBN|978-3-540-04507-6}}.</ref>  In fact, here <math>\frac{X}{X+Y}</math> and <math>X+Y</math> are independent, and <math>X+Y \sim \Gamma(\alpha + \beta, \theta)</math>. If <math>Z \sim \Gamma(\gamma, \theta)</math> and <math>Z</math> is independent of <math>X</math> and <math>Y</math>, then <math>\frac{X+Y}{X+Y+Z} \sim \Beta(\alpha+\beta,\gamma)</math> and <math>\frac{X+Y}{X+Y+Z}</math> is independent of <math>\frac{X}{X+Y}</math>. This shows that the product of independent <math>\Beta(\alpha,\beta)</math> and <math>\Beta(\alpha+\beta,\gamma)</math> random variables is a <math>\Beta(\alpha,\beta+\gamma)</math> random variable.

Also, the ''k''th [[order statistic]] of ''n'' [[Uniform distribution (continuous)|uniformly distributed]] variates is <math>\Beta(k, n+1-k)</math>, so an alternative if α and β are small integers is to generate α + β − 1 uniform variates and choose the α-th smallest.<ref name=David1/>

Another way to generate the Beta distribution is by [[Pólya urn model]]. According to this method, one start with an "urn" with α "black" balls and β "white" balls and draw uniformly with replacement. Every trial an additional ball is added according to the color of the last ball which was drawn. Asymptotically, the proportion of black and white balls will be distributed according to the Beta distribution, where each repetition of the experiment will produce a different value.

It is also possible to use the [[inverse transform sampling]].