Beta distribution as ratio involving Gamma distributions

Let $X$ and $Y$ be two random variables such that $X \sim \text{Gamma}(a, \lambda)$ and $Y \sim \text{Gamma}(b, \lambda)$. Let $W$ be a new random variable such that $W := \frac{X}{X + Y}$. If $X$ and $Y$ are independent then we can prove that $W \sim \text{Beta}(a, b)$. For example, we can compute the joint distribution of $W$ and $T := X + Y$ using the change of variables formula and then find the marginal distribution of $W$.

If $X$ and $Y$ are not independent however, what is the distribution of $W$? Is it still $\text{Beta}(a, b)$? If yes, how do I prove that $W \sim \text{Beta}(a, b)$? I tried conditioning on $Y$ and using the law of total probability, but I could not do much.

• In general, if they are not independent, the result no longer holds. Consider $X = Y$ almost surely, then $W = 1/2$ almost surely. If you want to compute the pdf in such case, you need to specify the joint pdf of $X, Y$ first. – BGM May 25 '18 at 3:35
• The answer to the question "does this result still hold if we drop the independence assumption" is almost always "no" with $X=Y$ as a counterexample. – spaceisdarkgreen May 25 '18 at 3:40
• @BGM What if I consider a non-extreme case $X \neq Y$? Suppose $a \neq b$. Can $W$ be then a Beta? – Supreeth Narasimhaswamy May 25 '18 at 5:28
• It is important to realize that distributions of $X$ and $Y$ say nothing about the joint distribution or distributions of functions of $X$ and $Y$ without independence except in very trivial cases. – Kabo Murphy May 25 '18 at 6:05
• The answer to the followup question “but what if $X\ne Y$?” is always “no, you have missed the point.” – spaceisdarkgreen May 25 '18 at 14:20