Expectation of a function of a normal distributed r.v.

Hallo everybody,

is there any chance to compute the following expectation explicitly? (I don't think so. Am I right?)

$\displaystyle F(a):=\mathbb{E}[(1+a(e^{Z-r}-1))^{\gamma}]$

where

$\displaystyle Z$ is a normally distributed r.v. $\displaystyle \mathcal{N}(\mu,\sigma^2)$

$\displaystyle a \in [0,1]$ a constant

$\displaystyle r,\gamma > 0 $ constants

Actually, I want to maximize the expectation w.r.t $\displaystyle a$ over the interval $\displaystyle [0,1]$ ($\displaystyle a$ is non-random). I know that the maximum exists since $\displaystyle F$ is concave and the interval compact. But does anyone has an idea how to compute the maximum?

Thanks in advance

Re: Expectation of a function of a normal distributed r.v.

is $\displaystyle \gamma$ an integer? if so you can expand your brackets and end up with a bunch of lognormal variables (the expectation of these is a standard result)

Re: Expectation of a function of a normal distributed r.v.

Thank you for your answer.

No, $\displaystyle \gamma$ is not an integer. In most of my applications $\displaystyle \gamma$ is between $\displaystyle 0$ and $\displaystyle 1.$

Re: Expectation of a function of a normal distributed r.v.

i dont know then, i suggest you lookup the derivation of the moments of a lognormal distribution since those are similar to what you need to do.