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

**Juju** · May 14th 2012, 01:30 AM

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?

Thanks in advance

**SpringFan25** · May 14th 2012, 01:55 AM

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)

**Juju** · May 14th 2012, 03:21 AM

Thank you for your answer.

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

**SpringFan25** · May 14th 2012, 06:58 AM

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.

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