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Math Help - 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?)

    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
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  2. #2
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    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)
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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, \gamma is not an integer. In most of my applications \gamma is between 0 and 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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