# Expected value of a non-linear function which involves a multinomial random variable

$\mathbf{X}$ is a random variable with a multinomial distribution $MN(n, p_1, \dots, p_k)$ (all parameters are known) and $\mathbf{q}=\{q_1,\dots,q_k\}$ is a fixed $k$-vector ($\mathbf{q}$ is not a constant vector to avoid a trivial case). Is there an exact analytical form or a nice and good approximation to $E_X\left(\frac{1}{1+\exp(\mathbf{X'}\mathbf{q})}\right)$?

This question comes from a model that I'm working on. My other solution will be to do a simulation of $\mathbf{X}$ to evaluate the expected value above instead. The expression seems pretty enough to guarantee more effort in trying to find a nicer way though.

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$$\sum\limits_{m\geqslant0}(-1)^m\cdot\left(\sum\limits_{\ell=1}^kp_\ell\,\mathrm e^{-(m+1)q_\ell}\right)^n$$