lognormal random variable

$X$ is a lognormal random variable with parameters $\mu \in ℝ$ and ${\sigma }^2>0$ if its probability density function is given for $x>0$ by

$f_X(x)={\displaystyle \frac{1}{\sqrt{2\pi {\sigma }^2}}}{\displaystyle \frac{e^{-\frac{{(\ln x-\mu )}^2}{2{\sigma }^2}}}{x}}.$

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 $\ln (X)$ 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.