distribution of the ratio of two gamma random variables

Question

Assume that $X \sim Ga(\alpha_1, \beta_1)$ and $Y \sim Ga(\alpha_2, \beta_2)$. Define $Z= X/Y$. What 's the distribution of $Z$?

Answer 1

$\beta_1X \sim Gamma(\alpha_1, 1)$ and $\beta_2 Y \sim Gamma(\alpha_2, 1)$, then according to Wikipedia

$$\dfrac{\beta_1X}{\beta_2Y} \sim \text{Beta Prime distribution}(\alpha_1, \alpha_2). $$

In addition, short hand you write $\beta'(\alpha_1, \alpha_2)$. Now the Wiki page also describes the density of the general Beta-prime distribution $\beta'(\alpha_1, \alpha_2, p, q)$, as

$$ f(x) = \dfrac{p \left( \frac{x}{q}\right)^{\alpha_1 p-1} \left(1 + \left(\frac{x}{q} \right)^p \right)^{-\alpha_1 -\alpha_2} }{q B(\alpha_1, \alpha_2)}.$$

The Beta-prime distribution is the special case of the general Beta-prime distribution when $p = q = 1$. In addition the wiki page says for a constant $k$

$$k\beta'(\alpha_1, \alpha_2, p, q) = \beta'(\alpha_1, \alpha_2, p, kq). $$

Thus, $$\dfrac{X}{Y} \sim \beta'\left(\alpha_1, \alpha_2, 1, \frac{\beta_2}{\beta_1} \right).$$

Answer 2

Greenparker's reply says it all, but you can also note that $\Gamma$ distributions can be expressed as scaled $\chi^2$ distributions, and that the properly scaled ratio of two $\chi^2$ random variables follows an $F$ distribution.

Following the parametrization and scaling rule above, if $X\sim\Gamma_{(\alpha,\beta)}$, then $2\beta X\sim\chi^2_{2\alpha}$

This means that for your two Gamma-distributed variables $X$ and $Y$,

$$\frac{\alpha_2\beta_1 X}{\alpha_1\beta_2 Y}\sim F_{(2\alpha_1,2\alpha_2)}$$

where $F$ is the $F$ distribution. You could also say that X/Y follows a scaled $F$ distribution.