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Suppose that $Y_1...,Y_n$ are independant gamma random variables: $Y_i \sim \Gamma(\alpha_i,\beta_i)$, with density $f_i(t) = \frac{t^{\alpha_i-1} e^{-\frac{x}{\beta}}}{\Gamma(\alpha) \beta^{\alpha}}$.

Suppose that $a_1,...a_m$ and $b_1,...,b_m$ are strictly positives reals.

I want to compute:

$$\mathbb E \left(\prod\limits_{j=1}^m \left(b_j + Y_1 +...+ Y_n\right)^{-a_j}\right)$$

My first attemp was expanding the function inside the expectation into a multivairate taylor serie in $(Y_1,...,Y_n)$ and then use the fact that $\mathbb E(Y_i^k)$ is known. But it is quite inneficient.

Maybe from the structure of the function we could expand in a 'power serie' with non-integer powers (which would probably depend on $a_1,...,a_m$), and the serie would converge more efficiently (since $\mathbb E(Y_i^a)$ is known for any $a$, even $a$ that are not integers).

How can i do that ?

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  • $\begingroup$ Is there a reason to expect a closed-form (or at least simpler) expression? Even when $n = m = 1$ the answer does not seem to be simple. $\endgroup$ Jan 15, 2021 at 17:53
  • $\begingroup$ For the record: $n = m = 1$ leads to some hypergeometric rubbish, and for $n = 1$ and $m = 2$ Mathematica is not able find a closed-form expression. $\endgroup$ Jan 15, 2021 at 18:00
  • $\begingroup$ I am pretty sure that a closed-form expression will not be posible, but a good convergent serie yes. For n=m=1, this expectation is in essence the laplace transform of the product of two gammas, which can be expressed by some confluent hypergeometric functions. For n,m > 1, i dont know yet but i have hope haha. $\endgroup$
    – lrnv
    Jan 15, 2021 at 18:02
  • $\begingroup$ Btw, could you share how you did it with mathematica ? $\endgroup$
    – lrnv
    Jan 15, 2021 at 18:07
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    $\begingroup$ By the way, I wonder why someone downvoted the question, it seems perfectly reasonable. $\endgroup$ Jan 15, 2021 at 18:12

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