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Thread: How to obtain moment generating function for lognormal distribution.

  1. #1
    Junior Member
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    How to obtain moment generating function for lognormal distribution.

    I can obtain
    $\displaystyle
    M_{X}(t) = e^{\mu t + \sigma^{2}t^{2}/2}
    $
    as the moment generating function of the normal distribution but I dont see the step to obtain
    $\displaystyle
    E(X^n) = e^{n \mu + \frac{1}{2}n^{2}\sigma^{2}}
    $
    as the Taylor series elements of the lognormal moment generating function.
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  2. #2
    MHF Contributor matheagle's Avatar
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    Since $\displaystyle Y=\ln X$ we have $\displaystyle e^Y=X$.

    So, $\displaystyle MGF_Y(t)=E(e^{Yt})=E(X^t)$ using $\displaystyle e^{Yt}=(e^Y)^t=X^t$.

    Or $\displaystyle MGF_Y(n)=E(X^n)$.
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