Let $X$ be a random variable from a normal distribution $N(\mu, \sigma)$. How do we calculate the expectation $E[e^{k\cdot e^{-X}}]$, where $k<0$?
I think we can use the moment generating function
$ E[e^{k\cdot e^{-X}}] = \sum_{t=0}^{\infty}\frac{k^{t}E[e^{-tX}]}{t!} $, where $E[e^{-tX}]=e^{-t\mu+\frac{1}{2}\sigma^{2}t^{2}}$
But is there a more "compact" solution beyond the one above?