I was looking at a proof of the following fact
Let $X \sim \mbox{Gamma}(\alpha, 1)$ and $Y \sim \mbox{Gamma}(\beta, 1)$ where the paramaterization is such that $\alpha$ is the shape parameter. Then
$$
\frac{X}{X + Y} \sim \mbox{Beta}(\alpha, \beta).
$$
found here in the second answer.
In the link I gave, when the author of the answer writes "To prove this, write the joint pdf $f_{X, Y} (x, y) = \frac{1}{\Gamma(\alpha) \Gamma(\beta)} x^{\alpha - 1} y^{\beta - 1} e^{-(x + y)}$ ", does this mean $X$ and $Y$ are to be considered independent?
Also the kind reader that may wish to give their proof of the above fact is very welcome to do so.
EDIT: when I copied pasted the linked answer I did not notice that the Lemma I would like to prove is stated a bit differently:
If $Z \sim Gamma(\alpha, \theta)$ and $W \sim Gamma(\beta, \theta)$, then $A = Z / (W+Z) \sim Beta(\alpha, \beta)$.
