Solved – Distribution of the quotient of two gamma random variables with different rate parameters

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 $c_1 neq c_2$?

If $X sim Gamma(a,1)$ is independent of $Y sim Gamma(b,1)$ then the ratio $X/Y$ has the Beta prime distribution with parameters $a$ and $b$.

In fact, the result holds if you replace the common value of the rate parameter ($1$ here) by any other value, because the rate parameter has this property: if $X sim Gamma(a,S)$, then $lambda times X sim Gamma(a, S/lambda)$ for any $lambda >0$.

Thus, $X' sim Gamma(a,c_1)$ is independent of $Y' sim Gamma(b,c_2)$, then the ratio $X'/Y'$ has the same distribution than $frac{c_2}{c_1} times X/Y$ where $X sim Gamma(a,1)$ is independent of $Y sim Gamma(b,1)$. Therefore, denoting by $f$ the pdf of the Beta prime distribution then the pdf of $X'/Y'$ is $r mapsto frac{c_1}{c_2} f(frac{c_1}{c_2} r)$. This scaled Beta prime distribution has no devoted name.

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