Log-normal

Log-normal

Probability density function Some log-normal density functions with identical parameter ${\displaystyle \mu }$ but differing parameters ${\displaystyle \sigma }$
Cumulative distribution function Cumulative distribution function of the log-normal distribution (with ${\displaystyle \mu =0}$ )
Notation	${\displaystyle \operatorname {Lognormal} (\mu ,\,\sigma ^{2})}$
Parameters	${\displaystyle \mu \in (-\infty ,+\infty )}$, ${\displaystyle \sigma >0}$
Support	${\displaystyle x\in (0,+\infty )}$
PDF	${\displaystyle {\frac {1}{x\sigma {\sqrt {2\pi }}}}\ e^{-{\frac {\left(\ln x-\mu \right)^{2}}{2\sigma ^{2}}}}}$
CDF	${\displaystyle {\frac {1}{2}}+{\frac {1}{2}}\operatorname {erf} {\Big [}{\frac {\ln x-\mu }{{\sqrt {2}}\sigma }}{\Big ]}}$
Mean	${\displaystyle \exp(\mu +\sigma ^{2}/2)}$
Median	${\displaystyle \exp(\mu )}$
Mode	${\displaystyle \exp(\mu -\sigma ^{2})}$
Variance	${\displaystyle [\exp(\sigma ^{2})-1]\exp(2\mu +\sigma ^{2})}$
Skewness	${\displaystyle (e^{\sigma ^{2}}\!\!+2){\sqrt {e^{\sigma ^{2}}\!\!-1}}}$
Ex. kurtosis	${\displaystyle \exp(4\sigma ^{2})+2\exp(3\sigma ^{2})+3\exp(2\sigma ^{2})-6}$
Entropy	${\displaystyle \log(\sigma e^{\mu +{\tfrac {1}{2}}}{\sqrt {2\pi }})}$
MGF	defined only for numbers with a non-positive real part, see text
CF	representation ${\displaystyle \sum _{n=0}^{\infty }{\frac {(it)^{n}}{n!}}e^{n\mu +n^{2}\sigma ^{2}/2}}$ is asymptotically divergent but sufficient for numerical purposes
Fisher information	${\displaystyle {\begin{pmatrix}1/\sigma ^{2}&0\\0&1/2\sigma ^{4}\end{pmatrix}}}$

In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable ${\displaystyle X}$ is log-normally distributed, then ${\displaystyle Y=\ln(X)}$ has a normal distribution. Likewise, if ${\displaystyle Y}$ has a normal distribution, then the exponential function of ${\displaystyle Y}$ is ${\displaystyle X=\exp(Y)}$ has a log-normal distribution. A random variable which is log-normally distributed takes only positive real values. The distribution is occasionally referred to as the Galton distribution or Galton's distribution, after Francis Galton. The log-normal distribution also has been associated with other names, such as McAlister, Gibrat and Cobb–Douglas.