If X1,X2X_1, X_2X1,X2 are i.i.d. random variables where X1∼Gamma(α1,β)X_1 \sim \text{Gamma}(\alpha_1, \beta)X1∼Gamma(α1,β) and X2∼Gamma(α2,β)X_2 \sim \text{Gamma}(\alpha_2, \beta)X2∼Gamma(α2,β), what is the distribution of Y=X1X1+X2Y = \frac{X_1}{X_1 + X_2}Y=X1+X2X1?
Beta(α1,α2)\text{Beta}(\alpha_1, \alpha_2)Beta(α1,α2)
Gamma(α1+α2,β)\text{Gamma}(\alpha_1+\alpha_2, \beta)Gamma(α1+α2,β)
Normal(0,1)\text{Normal}(0, 1)Normal(0,1)
Uniform(0,1)\text{Uniform}(0, 1)Uniform(0,1)