# Distribution of the ratio of Dirichlet/Gamma variates

It can be seen that the following random variates have the same distribution:

1. $$\frac{X_1 + X_3}{X_2 + X_3}$$, where $$(X_1, X_2, X_3) \sim \text{Dirichlet} (\alpha_1, \alpha_2, \alpha_3)$$
2. $$\frac{Y_1 + Y_3}{Y_2 + Y_3}$$, where $$(Y_0, Y_1, Y_2, Y_3) \sim \text{Dirichlet} (\alpha_0, \alpha_1, \alpha_2, \alpha_3)$$
3. $$\frac{Z_1 + Z_3}{Z_2 + Z_3}$$, where the $$(Z_i)_i$$ are independent and $$Z_i \sim \text{Gamma}(k = \alpha_i, \theta = 1)$$

Question: does this distribution have a name? Has it been studied somewhere in the literature? Were it not for $$X_1$$ in the numerator, it seems that this would be a Beta-Prime distribution.

• There is no $Y_0$ in your second formula. Commented Mar 11, 2023 at 15:31
• @StephanKolassa that's intentional. Commented Mar 11, 2023 at 22:35
• Once you know that Dirichlet distributions arise as ratios of Gamma variables, everything fits together. Such combinations of variables like $(X_1+X_3,X_2+X_3)$ have been used to formulate multivariate families of correlated Gamma distributions, suggesting one direction to search.
– whuber
Commented Mar 11, 2023 at 23:45
• @whuber thanks, unfortunately I'm not seeing how this gets me farther than the equivalence between 1., 2. and 3. Commented Mar 15, 2023 at 13:43
• @whuber thanks, I appreciate that. Commented Mar 17, 2023 at 20:46

I would personally suggest to call this a Dirichlet Process Mass Ratio distribution, denoted $$\text{DPMR}(\alpha_1, \alpha_2, \alpha_3)$$, reflecting the fact that such a random variable is the ratio of (random) probability masses given to two sets $$S_1$$ and $$S_2$$ by a Dirichlet Process. $$\alpha_1$$ corresponds to $$S_1-S_2$$, $$\alpha_2$$ to $$S_2-S_1$$, and $$\alpha_3$$ to $$S_1 \cap S_2$$, consistently with writing these indices in binary (1 -> 01, 2 -> 10, 3 -> 11).