Log-normal distribution

Log-normal distribution
	Probability density function; Identical parameter but differing parameters
	Cumulative distribution function;
Notation
Parameters	(logarithm of location), ; (logarithm of scale)
Support
PDF
CDF
Quantile	; ;
Mean
Median
Mode
Variance
Skewness
Excess kurtosis
Entropy
MGF	defined only for numbers with a; non-positive real part, see text
CF	representation ; is asymptotically divergent, but adequate; for most numerical purposes
Fisher information
Method of moments	; ;
Expected shortfall	; ;

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 $X$ is log-normally distributed, then $Y = ln(X)$ has a normal distribution.^[2]^[3] Equivalently, if $Y$ has a normal distribution, then the exponential function of $Y$ , $X = exp(Y)$ , has a log-normal distribution. A random variable which is log-normally distributed takes only positive real values. It is a convenient and useful model for measurements in exact and engineering sciences, as well as medicine, economics and other topics (e.g., energies, concentrations, lengths, prices of financial instruments, and other metrics).

The distribution is occasionally referred to as the Galton distribution or Galton's distribution, after Francis Galton.^[4] The log-normal distribution has also been associated with other names, such as McAlister, Gibrat and Cobb–Douglas.^[4]

A log-normal process is the statistical realization of the multiplicative product of many independent random variables, each of which is positive. This is justified by considering the central limit theorem in the log domain (sometimes called Gibrat's law). The log-normal distribution is the maximum entropy probability distribution for a random variate $X$ —for which the mean and variance of $ln(X)$ are specified.^[5]

^ Norton, Matthew; Khokhlov, Valentyn; Uryasev, Stan (2019). "Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation" (PDF). Annals of Operations Research. 299 (1–2). Springer: 1281–1315. arXiv:1811.11301. doi:10.1007/s10479-019-03373-1. S2CID 254231768. Archived (PDF) from the original on 2021-04-18. Retrieved 2023-02-27 – via stonybrook.edu.
^ Weisstein, Eric W. "Log Normal Distribution". mathworld.wolfram.com. Retrieved 2020-09-13.
^ "1.3.6.6.9. Lognormal Distribution". www.itl.nist.gov. U.S. National Institute of Standards and Technology (NIST). Retrieved 2020-09-13.
^ ^a ^b Cite error: The named reference JKB was invoked but never defined (see the help page).
^ Park, Sung Y.; Bera, Anil K. (2009). "Maximum entropy autoregressive conditional heteroskedasticity model" (PDF). Journal of Econometrics. 150 (2): 219–230, esp. Table 1, p. 221. CiteSeerX 10.1.1.511.9750. doi:10.1016/j.jeconom.2008.12.014. Archived from the original (PDF) on 2016-03-07. Retrieved 2011-06-02.

[norton-1] Norton, Matthew; Khokhlov, Valentyn; Uryasev, Stan (2019). "Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation" (PDF). Annals of Operations Research. 299 (1–2). Springer: 1281–1315. arXiv:1811.11301. doi:10.1007/s10479-019-03373-1. S2CID 254231768. Archived (PDF) from the original on 2021-04-18. Retrieved 2023-02-27 – via stonybrook.edu.

[:1-2] Weisstein, Eric W. "Log Normal Distribution". mathworld.wolfram.com. Retrieved 2020-09-13.

[:2-3] "1.3.6.6.9. Lognormal Distribution". www.itl.nist.gov. U.S. National Institute of Standards and Technology (NIST). Retrieved 2020-09-13.

[JKB-4] Cite error: The named reference JKB was invoked but never defined (see the help page).

[5] Park, Sung Y.; Bera, Anil K. (2009). "Maximum entropy autoregressive conditional heteroskedasticity model" (PDF). Journal of Econometrics. 150 (2): 219–230, esp. Table 1, p. 221. CiteSeerX 10.1.1.511.9750. doi:10.1016/j.jeconom.2008.12.014. Archived from the original (PDF) on 2016-03-07. Retrieved 2011-06-02.

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Probability density function Identical parameter $\ \mu \$ but differing parameters $\sigma$
Cumulative distribution function $\ \mu =0\$
Notation	$\ \operatorname {Lognormal} \left(\ \mu ,\,\sigma ^{2}\ \right)\$
Parameters	$\ \mu \in (\ -\infty ,+\infty \ )\$ (logarithm of location), $\ \sigma >0\$ (logarithm of scale)
Support	$\ x\in (\ 0,+\infty \ )\$
PDF	$\ {\frac {1}{\ x\sigma {\sqrt {2\pi \ }}\ }}\ \exp \left(-{\frac {\left(\ln x-\mu \ \right)^{2}}{2\sigma ^{2}}}\right)$
CDF	$\ {\frac {\ 1\ }{2}}\left[1+\operatorname {erf} \left({\frac {\ \ln x-\mu \ }{\sigma {\sqrt {2\ }}}}\right)\right]=\Phi \left({\frac {\ln(x)-\mu }{\sigma }}\right)$
Quantile	$\ \exp \left(\mu +{\sqrt {2\sigma ^{2}}}\operatorname {erf} ^{-1}(2p-1)\right)\$ $=\exp(\mu +\sigma \Phi ^{-1}(p))$
Mean	$\ \exp \left(\ \mu +{\frac {\sigma ^{2}}{2}}\ \right)\$
Median	$\ \exp(\ \mu \ )\$
Mode	$\ \exp \left(\ \mu -\sigma ^{2}\ \right)\$
Variance	$\ \left[\ \exp(\sigma ^{2})-1\ \right]\ \exp \left(2\ \mu +\sigma ^{2}\right)\$
Skewness	$\ \left[\ \exp \left(\sigma ^{2}\right)+2\ \right]{\sqrt {\exp(\sigma ^{2})-1\;}}$
Excess kurtosis	$\ 1\ \exp \left(4\ \sigma ^{2}\right)+2\ \exp \left(3\ \sigma ^{2}\right)+3\ \exp \left(2\sigma ^{2}\right)-6\$
Entropy	$\ \log _{2}\left(\ {\sqrt {2\pi \ }}\ \sigma \ e^{\mu +{\tfrac {1}{2}}}\ \right)\$
MGF	defined only for numbers with a non-positive real part, see text
CF	representation $\ \sum _{n=0}^{\infty }{\frac {\ (i\ t)^{n}\ }{n!}}e^{\ n\mu +n^{2}\sigma ^{2}/2}\$ is asymptotically divergent, but adequate for most numerical purposes
Fisher information	$\ {\begin{pmatrix}{\frac {1}{\ \sigma ^{2}\ }}&0\\0&{\frac {2}{\ \sigma ^{2}\ }}\end{pmatrix}}\$
Method of moments	$\ \mu =\log \left({\frac {\operatorname {\mathbb {E} } [X]\ }{\ {\sqrt {{\frac {\ \operatorname {Var} [X]~~}{\ \operatorname {\mathbb {E} } [X]^{2}\ }}+1\ }}\ }}\right)\ ,$ $\ \sigma ={\sqrt {\log \left({\frac {\ \operatorname {Var} [X]~~}{\ \operatorname {\mathbb {E} } [X]^{2}\ }}+1\ \right)\ }}$
Expected shortfall	$\ {\frac {1}{2}}e^{\mu +{\frac {\sigma ^{2}}{2}}}{\frac {\ 1+\operatorname {erf} \left({\frac {\sigma }{\ {\sqrt {2\ }}\ }}+\operatorname {erf} ^{-1}(2p-1)\right)\ }{p}}$ ^[1] $\ =e^{\mu +{\frac {\sigma ^{2}}{2}}}{\frac {1}{1-p}}(1-\Phi (\Phi ^{-1}(p)-\sigma ))$