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

  1. #1
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    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
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  2. #2
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    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)
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  3. #3
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    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 $\displaystyle 1.$
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  4. #4
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    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.
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