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

That means, given X=Log(Y) with

$\displaystyle

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

$\displaystyle

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!