# 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. 1.

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

2. 2.

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

3. 3.

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

4. 4.

$M_{X}(t)$ is not a useful quantity.

Title lognormal random variable LognormalRandomVariable 2013-03-22 11:54:46 2013-03-22 11:54:46 mathcam (2727) mathcam (2727) 12 mathcam (2727) Definition msc 62E15 lognormal distribution