Mean of Generalization of the Dirichlet Distribution 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!

Fair warning- we will not get far. This is not even close to a full answer, but I think it is a bit more than a comment.
As mentioned by Matt F. in the comments, there is highly unlikely to be a closed-form solution. To see why, consider that the denominator's density itself "barely" has a closed-form, with a density given by the infinite series set out in equation 2.9 in Moschopoulos (1985) https://doi.org/10.1007/BF02481056.
Specifically, the sum of n gamma distributions is given by:
$$
\begin{aligned}
g(y)=C\sum_{k=0}^\infty\frac{\delta_k y^{\rho+k-1}e^{-y/{\beta_1}}}{\Gamma(\rho+k)\beta_1^{\rho+k}}\\
\end{aligned}
$$
Here, $\{\delta_k\}$ is an infinite series of constants, $\rho$ and $C$ are also constants, and $\beta_1$ is the smallest value of $\theta_i$
Because the support is positive, you can follow the procedure here to get the density of the reciprocal distribution of the denominator https://en.wikipedia.org/wiki/Inverse_distribution. But the dependence between the numerator and the denominator (and it's reciprocal) seems to preclude a "nice" answer, even with the moments.
So, obviously getting the moments numerically is not so bad. But a closed-form seems unlikely given the above.
