
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
$\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!

Resolved...
The transformation theorem is based upon several assumptions (monotonicity, especially) that are valid for the pdf in exam.
Shortly and basically it says:
$\displaystyle
f_Y(y) = f_X[h(y)]\left \frac{d\ h(y)}{dy}\right
$
Hence:
$\displaystyle
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.