Lognormal PDF

**paolopiace** · October 4th 2008, 11:50 AM

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

That means, given X=Log(Y) with

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

Well, I’m 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!

**paolopiace** · October 4th 2008, 03:45 PM

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

Shortly and basically it says:

$ f_Y(y) = f_X[h(y)]\left| \frac{d\ h(y)}{dy}\right| $

Hence:

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

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

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