Let $X \in \text{Gamma}(\alpha, 1)$ and $X \in \text{Gamma}(\beta, 1)$ each be a D3838: Gamma random positive real number such that

(i)	$X, Y$ is an D2713: Independent random collection

A D5722: Random positive real number $B \in \text{Random}(0, 1)$ is a beta random positive real number with parameter $(\alpha, \beta)$ if and only if \begin{equation} B \overset{d}{=} \frac{X}{X + Y} \end{equation}