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Math Help - Lognormal PDF

  1. #1
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    Lognormal PDF

    Definition: a random variable Y is defined as Lognormal distributed if its logarithm is Normal distributed.

    That means, given X=Log(Y) with

    <br />
X \sim N(\mu,\sigma)\ \mbox{ i.e. }\<br />
f_X = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}<br />
    then
    <br />
Y \sim \mbox{Log-}N(\mu,\sigma)\ \mbox{ i.e. }\<br />
f_Y = \frac{1}{y\sigma\sqrt{2\pi}} e^{-\frac{1}{2}(\frac{\log(y)-\mu}{\sigma})^2}<br />

    Well, Iím having very hard time to prove this. How can I go from the PDF of X to the pdf of Y?

    In a previous post, Tukeywilliams says "use the transformation theorem to get the pdf of the lognormal". I tried unsuccesfully...

    I would appreciate any help. Thanks!
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  2. #2
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    Resolved...

    The transformation theorem is based upon several assumptions (monotonicity, especially) that are valid for the pdf in exam.

    Shortly and basically it says:

    <br />
f_Y(y) = f_X[h(y)]\left| \frac{d\ h(y)}{dy}\right|<br />

    Hence:

    <br />
f_Y(y) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2} (\frac{\log y -\mu}{\sigma})^2} \left|\frac{1}{y}\right|<br />

    where h(y) = log y, hence y>0.
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