Suitable distribution to make a problem tractable

Question

Consider a set of random variables $x_i$ for $i\in\{1,\dots,n\}$. They are all drawn in an iid way from some distribution $F$. Now consider the function $$ Y(x)=A\log\left(\sum_{i}B_{i}x_{i}^{a}\right)+C\log\left(\sum_{i}D_{i}x_{i}^{b}\right)+E $$ where $a>0$, $b>0$, $A>0$, $B_i\geq 0$, $C>0$, $D_i\geq 0$ and $E>0$. I would like to find a suitable distribution $F$ so that I can have a closed-form solution for the distribution of $Y$. The simpler $Y$ is the better! Any suggestions would be much appreciated.

Answer 1

Make $x_i$ deterministic. That is the simplest possible $Y$.