Log-normal distribution

Log-normal
Probability density function Some log-normal density functions with identical location parameter $\mu$ $\mu$ but differing scale parameters $\sigma$ $\sigma$
Cumulative distribution function Cumulative distribution function of the log-normal distribution (with $\mu =0$ $\mu =0$ )
Notation	$\ln {\mathcal {N}}(\mu ,\,\sigma ^{2})$ $\ln {\mathcal {N}}(\mu ,\,\sigma ^{2})$
Parameters	$\mu \in \mathbb {R}$ $\mu \in \mathbb {R}$ — location, $\sigma >0$ $\sigma >0$ — scale of associated normal
Support	$x\in (0,+\infty )$ $x\in (0,+\infty )$
PDF	${\frac {1}{x\sigma {\sqrt {2\pi }}}}\ e^{-{\frac {\left(\ln x-\mu \right)^{2}}{2\sigma ^{2}}}}$ ${\frac {1}{x\sigma {\sqrt {2\pi }}}}\ e^{-{\frac {\left(\ln x-\mu \right)^{2}}{2\sigma ^{2}}}}$
CDF	${\frac {1}{2}}+{\frac {1}{2}}\,\mathrm {erf} {\Big [}{\frac {\ln x-\mu }{{\sqrt {2}}\sigma }}{\Big ]}$ ${\frac {1}{2}}+{\frac {1}{2}}\,\mathrm {erf} {\Big [}{\frac {\ln x-\mu }{{\sqrt {2}}\sigma }}{\Big ]}$
Mean	$e^{\mu +\sigma ^{2}/2}$ $e^{\mu +\sigma ^{2}/2}$
Median	$e^{\mu }\,$ $e^{\mu }\,$
Mode	$e^{\mu -\sigma ^{2}}$ $e^{\mu -\sigma ^{2}}$
Variance	$(e^{\sigma ^{2}}\!\!-1)e^{2\mu +\sigma ^{2}}$ $(e^{\sigma ^{2}}\!\!-1)e^{2\mu +\sigma ^{2}}$
Skewness	$(e^{\sigma ^{2}}\!\!+2){\sqrt {e^{\sigma ^{2}}\!\!-1}}$ $(e^{\sigma ^{2}}\!\!+2){\sqrt {e^{\sigma ^{2}}\!\!-1}}$
Ex. kurtosis	$e^{4\sigma ^{2}}\!\!+2e^{3\sigma ^{2}}\!\!+3e^{2\sigma ^{2}}\!\!-6$ $e^{4\sigma ^{2}}\!\!+2e^{3\sigma ^{2}}\!\!+3e^{2\sigma ^{2}}\!\!-6$
Entropy	$\log(\sigma e^{\mu +{\tfrac {1}{2}}}{\sqrt {2\pi }})$ $\log(\sigma e^{\mu +{\tfrac {1}{2}}}{\sqrt {2\pi }})$
MGF	defined only on the negative half-axis, see text
CF	representation $\sum _{n=0}^{\infty }{\frac {(it)^{n}}{n!}}e^{n\mu +n^{2}\sigma ^{2}/2}$ $\sum _{n=0}^{\infty }{\frac {(it)^{n}}{n!}}e^{n\mu +n^{2}\sigma ^{2}/2}$ is asymptotically divergent but sufficient for numerical purposes
Fisher information	${\begin{pmatrix}1/\sigma ^{2}&0\\0&2/\sigma ^{2}\end{pmatrix}}$ ${\begin{pmatrix}1/\sigma ^{2}&0\\0&2/\sigma ^{2}\end{pmatrix}}$

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 $X$ $X$ is log-normally distributed, then $Y=\ln(X)$ $Y=\ln(X)$ has a normal distribution. Likewise, if $Y$ $Y$ has a normal distribution, then $X=\exp(Y)$ $X=\exp(Y)$ 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.

...
Wikipedia