What is distribution of ratio of two gamma function Let suppose there are two gamma function X and Y.
$$X:Gamma(10,\alpha)$$
$$Y:Gamma(10,\beta)$$
I found that if  $\alpha$ and $\beta$ are same, the distribution is beta distribution.
But i like to know case when second parameter is different.
How can i found it? 
 A: Here is a paper on this topic: Malik, 1967 
"Exact Distribution of the Quotient of Independent Generalized Gamma Variables"
According to Rao and Garg, 1969
"It may be called a generalization of the beta distribution of the second kind..."
A: Fact 1. Let $X \sim \Gamma\left(n, \alpha\right)$, and let $Y = \gamma X$ for a positive constant $\gamma$, then $Y \sim X \sim \Gamma\left(n, \gamma \alpha\right)$.
Fact 2. Let $X \sim \Gamma\left(n, 1\right)$, $Y \sim \Gamma\left(n, 1\right)$ be independent random variables. Ratio of these variables, $Z = \frac{X}{Y}$ has beta-prime distribution with $\alpha=\beta=n$, and pdf
$$
    f_Z\left(z\right) = \frac{1}{\mathrm{B}\left(n, n\right)} \cdot \frac{z^{n-1}}{\left(1+z\right)^{2 n}} \cdot [ z > 0 ]
$$ 
where $[ \cdot ]$ denotes Iverson bracket, and $\mathrm{B}\left(a, b\right)$ denotes Euler's beta-function.
Combining these facts, for $X_a \sim \Gamma\left(n, a\right)$, $Y_b \sim \Gamma\left(n, b\right)$:
$$
   Z_{ab} = \frac{X_a}{Y_b} = \frac{a}{b} \frac{X}{Y} = \frac{a}{b} Z
$$
