How to obtain moment generating function for lognormal distribution.

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May 12th 2009, 09:50 PM
bob000

How to obtain moment generating function for lognormal distribution.

I can obtain
$<br /> M_{X}(t) = e^{\mu t + \sigma^{2}t^{2}/2}<br />$
as the moment generating function of the normal distribution but I dont see the step to obtain
$<br /> E(X^n) = e^{n \mu + \frac{1}{2}n^{2}\sigma^{2}}<br />$
as the Taylor series elements of the lognormal moment generating function.
May 12th 2009, 10:50 PM
matheagle

Since $Y=\ln X$ we have $e^Y=X$ .

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

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

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