5.6 … para proporções
5.6.1 Beta \(\cdot\) \(Beta(\alpha,\beta)\)
5.6.2 Dirichlet \(\cdot\) \(Dir(\alpha_1,\ldots,\alpha_k)\)
\(0 \le x_i \le 1\), \(\alpha_i > 0\), \(i \in \{1,\ldots,k\}\). \[\begin{equation} f(x_1,\ldots,x_k|\alpha_1,\ldots,\alpha_k) = \frac{\Gamma(\sum_{i=1}^{k} \alpha_i)}{\prod _{i=1}^{k}\Gamma(\alpha_i)} \prod_{i=1}^{k} x_i^{\alpha_i-1} \tag{5.9} \end{equation}\]
\[\begin{equation} E(X_i) = \frac{\alpha_i}{\sum_{j=1}^{k} \alpha_j} \tag{5.10} \end{equation}\]
\[\begin{equation} V(X_i) = \frac{\frac{\alpha_i}{\sum_{j=1}^k \alpha_j} \left( 1- \frac{\alpha_i}{\sum_{j=1}^k \alpha_j} \right)}{\sum_{j=1}^k \alpha_j + 1} \tag{5.11} \end{equation}\]