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

• May 14th 2012, 02:30 AM
Juju
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?)

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

where
$Z$ is a normally distributed r.v. $\mathcal{N}(\mu,\sigma^2)$
$a \in [0,1]$ a constant
$r,\gamma > 0$ constants

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

• May 14th 2012, 02:55 AM
SpringFan25
Re: Expectation of a function of a normal distributed r.v.
is $\gamma$ 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)
• May 14th 2012, 04:21 AM
Juju
Re: Expectation of a function of a normal distributed r.v.
No, $\gamma$ is not an integer. In most of my applications $\gamma$ is between $0$ and $1.$