I have a question about how to derive the distribution of the quotient of two random gamma variables drawn from two different Gamma distributions with the same shape, but different rates. For example, given $$ \theta \sim \frac{1}{Gamma(a, c_1)} \\ \tau \sim \frac{1}{Gamma(b, c_2)} $$ How do I find the distribution of the following? $$ M = \frac{\tau}{\sigma + \tau} $$
I've seen online that it would be $$M \sim \frac{1}{Beta(a+b,c)}$$ if $c_1 = c_2$. But what if $c_1 \neq c_2$?