Dirichlet distribution

Example

Let $K \ge 2$ and $\alpha_1, \dots, \alpha_K > 0$, and let $\alpha_0 = \alpha_1 + \dots + \alpha_K$. The Dirichlet distribution $\operatorname{Dir}(\alpha_1, \dots, \alpha_K)$ is the law of a random vector $(X_1, \dots, X_K)$ in the open simplex $\set{x \in \R^K : x_i > 0, \ x_1 + \dots + x_K = 1}$. The vector $(X_1, \dots, X_{K-1})$ has the density $$f(x_1, \dots, x_{K-1}) = \frac{\Gamma(\alpha_0)}{\Gamma(\alpha_1) \cdots \Gamma(\alpha_K)} \prod_{i=1}^K x_i^{\alpha_i - 1}$$ on the set where all $x_i > 0$ and $x_1 + \dots + x_{K-1} < 1$, with $x_K = 1 - x_1 - \dots - x_{K-1}$; here $\Gamma$ is the gamma function. A construction: if $Y_1, \dots, Y_K$ are independent and $Y_i$ has the gamma distribution with shape $\alpha_i$ and scale $1$, then $(Y_1, \dots, Y_K) / (Y_1 + \dots + Y_K)$ has law $\operatorname{Dir}(\alpha_1, \dots, \alpha_K)$. Check for $K = 2$: the density becomes $\frac{\Gamma(\alpha_1 + \alpha_2)}{\Gamma(\alpha_1)\Gamma(\alpha_2)} x^{\alpha_1 - 1} (1 - x)^{\alpha_2 - 1}$ on $(0, 1)$, so $X_1$ has the beta distribution with parameters $\alpha_1, \alpha_2$. The means are $E[X_i] = \alpha_i / \alpha_0$ and the variances are $\operatorname{Var}(X_i) = \frac{\alpha_i (\alpha_0 - \alpha_i)}{\alpha_0^2 (\alpha_0 + 1)}$.