\begin{definition}[Definition:Beta Distribution]
Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.
Let $\Img X = \closedint 0 1$.
$X$ is said to have a '''Beta distribution''' {{iff}} it has probability density function: 
:$\map {f_X} X = \dfrac {x^{\alpha - 1} \paren {1 - x}^{\beta - 1} } {\map \Beta {\alpha, \beta} }$
for $\alpha, \beta > 0$, where $\Beta$ denotes the Beta function.
This is written: 
:$X \sim \BetaDist \alpha \beta$
\end{definition}