Definition

$X \sim Beta (α, β)$ where $α, β \in R^{+}$ are the shape parameters

Properties

$f (x) = \frac{Γ ( α + β )}{Γ ( α ) Γ ( β )} x^{α - 1} (1 - x)^{β - 1} \in (0, 1)$

$E (X) = \frac{α}{α + β}$

$Var (X) = \frac{α β}{( α + β + 1 ) ( α + β ) ^{2}}$

Let $X_{1} \sim Γ (α, 1), X_{2} \sim Γ (β, 1)$ are independent gamma distributions, then $\frac{X _{1}}{X _{1} + X _{2}} \sim Beta (α, β)$