# Ratio of two beta random variables

I'm working on a problem for an hour and I wanted to get some hints. Suppose:

$y_1, y_2, y_3, y_4 \sim Dir(\alpha_1, \alpha_2, \alpha_3, \alpha_4)$

what is the distribution of $\frac{y_1}{y_1 + y_2}$ ?

My guess is that distribution should be $Beta(\alpha_1, \alpha_2)$

Could you guys give me some hints on how to show it?

• What is the beta distribution specified by four parameters? Commented May 7, 2015 at 17:26
• @HowDoIMath, my bad! it's dirichlet. Commented May 7, 2015 at 17:35

By the story of the dirichlet distribution, given $$X_{1} \sim \Gam(\alpha_{1}, 1)$$, it follows that the distribution of $$y_{1}$$ can be expressed as: \begin{equation*}\begin{aligned} y_{1} &\sim \frac{X_{1}}{\sum_{i = 1}^{4} X_{i}} \end{aligned}\end{equation*}
Therefore, we can write the distribution of $$\alpha = \frac{y_{1}}{y_{1} + y_{2}}$$ as: \begin{equation*}\begin{aligned} \alpha &\sim \frac{y_{1}}{y_{1} + y_{2}}\\ &\sim \frac{X_{1}/\sum_{i = 1}^{4} X_{i}}{X_{1}/\sum_{i = 1}^{4} X_{i} + X_{2}/\sum_{i = 1}^{4} X_{i}}\\ &\sim \frac{X_{1}}{X_{1} + X_{2}}\\ &\sim \Dir(\alpha_{1}, \alpha_{2})\\ &\sim \Beta(\alpha_{1}, \alpha_{2}) \end{aligned}\end{equation*}
Which shows that $$\alpha \sim \Beta(\alpha_{1}, \alpha_{2})$$.