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Let \(\alpha,\beta>0\) and let X be a random variable with p.d.f. \[f(x|\alpha,\beta)=\begin{cases} \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}x^{\alpha - 1}(1-x)^{\beta - 1}\quad \mathrm{for}\, 0<x<1,\\ 0\quad \mathrm{otherwise} \end{cases}\] The X has the beta distribution with parameters \(\alpha\) and \(\beta\).