lognormal random variable
$X$ is a lognormal random variable with parameters $\mu \in \mathbb{R}$ and ${\sigma}^{2}>0$ if its probability density function^{} is given for $x>0$ by
${f}_{X}(x)={\displaystyle \frac{1}{\sqrt{2\pi {\sigma}^{2}}}}{\displaystyle \frac{{e}^{\frac{{(\mathrm{ln}x\mu )}^{2}}{2{\sigma}^{2}}}}{x}}.$ 
To denote this, one usually writes $X\sim LogN(\mu ,{\sigma}^{2})$.
For a lognormal random variable $X$:

1.
$X$ is a random variable^{} such that $\mathrm{ln}(X)$ is a normal random variable with mean $\mu $ and variance^{} ${\sigma}^{2}$.

2.
$E[X]={e}^{\mu +{\sigma}^{2}/2}$

3.
$Var[X]={e}^{2\mu +{\sigma}^{2}}({e}^{{\sigma}^{2}}1)$

4.
${M}_{X}(t)$ is not a useful quantity.
Title  lognormal random variable 

Canonical name  LognormalRandomVariable 
Date of creation  20130322 11:54:46 
Last modified on  20130322 11:54:46 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  12 
Author  mathcam (2727) 
Entry type  Definition 
Classification  msc 62E15 
Synonym  lognormal distribution 