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

• May 14th 2012, 01:30 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?)

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

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?

• May 14th 2012, 01:55 AM
SpringFan25
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)
• May 14th 2012, 03:21 AM
Juju
Re: Expectation of a function of a normal distributed r.v.
No, $\displaystyle \gamma$ is not an integer. In most of my applications $\displaystyle \gamma$ is between $\displaystyle 0$ and $\displaystyle 1.$