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

\displaystyle f_{X}(x)=\frac{1}{\sqrt{2\pi\sigma^{2}}}\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.