If $X \sim \text{exponential}(1)$
I need to proof, using transformation method, that
$Y = \dfrac{\sum_{i=1}^{\alpha}X_i}{\sum_{i=1}^{\alpha+\beta}X_i} \sim Beta(\alpha, \beta)$
I know that $\sum_{i=1}^{\alpha}X_i\sim$ Gamma$(\alpha, 1)$ and $\sum_{i=1}^{\alpha+\beta}X_i\sim$ Gamma$(\alpha+\beta, 1)$, but I can't proof that Y follows Beta using transformation method. Can anyone help?
