I am trying to show the following:
\begin{equation*} E[e^{-\gamma W}]=e^{-\gamma(E[W]-\frac{\gamma}{2}Var [W])} \end{equation*}
but I really can't remember what I am supposed to do to get from the LHS to the RHS. I have tried using integration this way
\begin{equation*} \int We^{-\gamma W}dW \end{equation*}
and then use integration by parts, but even though what I get resembles it, it can't be correct (because $e^{-\gamma W}$ is not the distribution of W).
I have also tried using Taylor series expansion, but I think I am way off, and I don't think an approximation here is what I need, because the equality above is exact.
FYI, this is not homework, I am working through a paper (page 10) and I would really like to know how every step was derived.
Can anyone at least point me to the right direction?
EDIT: This expectation on the RHS is very similar to the moment generating function formula (with a negative exponent). If you check here, you will see that the moment generating function for the normal distribution is like the LHS (but with a positive sign). So in a way I have my answer, but I still would like to know how to derive it, if there is a way. I know little if anything at all about moment generating functions, so maybe I shouldn't try and derive it but rather just use the result? Does it even make sense to try and derive it?