#StackBounty: #gamma-distribution #dirichlet-distribution #multivariate-distribution #generalization #multinomial-dirichlet-distributio…

Bounty: 100

I know that if $X_{1},X_{2},…X_{n}$ are independent $mathrm{Gamma}(alpha_{i},theta)$ – distributed variables (notice they all have the same scale parameter $theta$) and

$Y_{i}=frac{X_{i}}{sum_{j=1}^{n}X_{j}}$

then :

$Y=(Y_{1},Y_{2},…Y_{n});$~$;mathrm{Dirichlet}(alpha_1,alpha_2,…,alpha_n)$

I’m interested in what happens if $X_{1},X_{2},…X_{n}$ are allowed to have different scale parameters. That is:

$X_i$~$mathrm{Gamma}(alpha_i,theta_i)$

The problem is: find a closed-form solution for the expectation of $Y_i ; forall i$.

Great if you can also tell me how this distribution is called and provide a reference (Textbook, paper).

Bounty’s on. Thank you!


Get this bounty!!!

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