Beta

The Beta distribution is defined for 0≤x≤1 by the integral

$\displaystyle \limfunc$BetaDist$\displaystyle \left(\vphantom{ x;v,w}\right.$x;v, w$\displaystyle \left.\vphantom{ x;v,w}\right)$ = $\displaystyle {\frac{{1}}{{B\left( v,w\right) }%
}}$$\displaystyle \int_{{0}}^{{x}}$uv-1$\displaystyle \left(\vphantom{ 1-u}\right.$1 - u$\displaystyle \left.\vphantom{ 1-u}\right)^{{w-1}}_{}$du

where B(v, w) = $\int_{{0}}^{{1}}$uv-1$\left(\vphantom{ 1-u}\right.$1 - u$\left.\vphantom{ 1-u}\right)^{{w-1}}_{}$du is the Beta function with parameters v and w.

The probability density function for the beta distribution is

$\displaystyle \limfunc$BetaDen$\displaystyle \left(\vphantom{ u;v,w}\right.$u;v, w$\displaystyle \left.\vphantom{ u;v,w}\right)$ = $\displaystyle {\frac{{u^{v-1}\left( 1-u\right) ^{w-1}}}{{%
B\left( v,w\right) }}}$

The parameters v and w are positive real numbers called shape parameters, and 0≤u≤1. The mean of the beta distribution is ${\dfrac{{v}}{{v+w}}}$.

$\blacktriangleright$ Evaluate Numerically

$\limfunc$BetaDist$\left(\vphantom{ .5;2,3}\right.$.5;2, 3$\left.\vphantom{ .5;2,3}\right)$ =  .6875

$\limfunc$BetaDen$\left(\vphantom{ .5;2,3}\right.$.5;2, 3$\left.\vphantom{ .5;2,3}\right)$ =  1.5

The following plots show cumulative distribution functions $\limfunc$BetaDist$\left(\vphantom{ x;b,c}\right.$x;b, c$\left.\vphantom{ x;b,c}\right)$ and probability density functions $\limfunc$BetaDen(x;b, c) for (b, c) = $\left(\vphantom{ 2,3}\right.$2, 3$\left.\vphantom{ 2,3}\right)$,  $\left(\vphantom{ 5,1}\right.$5, 1$\left.\vphantom{ 5,1}\right)$,  $\left(\vphantom{
3,8}\right.$3, 8$\left.\vphantom{
3,8}\right)$, and 0≤x≤1.

itbpFU2.1577in1.7573in0inBeta density functions itbpFU2.1577in1.7573in0inBeta distribution functions