Generalized Dirichlet Distribution - General Moment Function

General Moment Function

If, then

E\left= \prod_{j=1}^k \frac{ \Gamma\left(\alpha_j+\beta_j\right) \Gamma\left(\alpha_j+r_j\right) \Gamma\left(\beta_j+\delta_j\right) }{ \Gamma\left(\alpha_j\right) \Gamma\left(\beta_j\right) \Gamma\left(\alpha_j+\beta_j+r_j+\delta_j\right) }

where for and . Thus

E\left(X_j\right)=\frac{\alpha_j}{\alpha_j+\beta_j}\prod_{m=1}^{j-1}\frac{\beta_m}{\alpha_m+\beta_m}.

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