Is there any closed form expression for $E(X e^{- \mu \sqrt{X}})$, where $X\sim Poisson(\lambda)$ and $\mu >0$? If not, is there any tight upper bound for this quantity? Any idea how to proceed?
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$\begingroup$ "tight upper bound" for which regime? E.g., $\lambda\to \infty$ and/or $\mu \to \infty$, etc.... $\endgroup$– Serguei PopovCommented Apr 7, 2016 at 22:33
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$\begingroup$ ideally for all possible $\lambda$ and $\mu$...but for me the more interesting regime would be $\mu > \lambda$, but I am not sure if this would help in finding a tight upper bound. $\endgroup$– UditaCommented Apr 8, 2016 at 6:09
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$\begingroup$ Asking for too good asymptotics on too much of a multidimensional domain makes it much harder to answer the question. $\endgroup$– Douglas ZareCommented Apr 9, 2016 at 20:57
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1 Answer
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Split ?
$$E(Xe^{-\mu\sqrt{X}})=E(Xe^{-\sqrt{\mu^2X}})\leq E(Xe^{-\mu^2X}\mathbf{1}_{\mu^2X\leq 1})+E(X\mathbf{1}_{\mu^2X\geq 1})$$
