Editing Log-normal distribution (section)

{{Short description|Probability distribution}}
{{Infobox probability distribution
 | name = Log-normal distribution
 | type = continuous
 | pdf_image = [[Image:Log-normal-pdfs.png|300px|Plot of the Lognormal PDF]]<br/><small>Identical parameter <math> \mu </math> but differing parameters <math>\sigma</math></small>
 | cdf_image = [[Image:Log-normal-cdfs.png|300px|Plot of the Lognormal CDF]]<br/><small><math> \mu = 0 </math></small>
 | notation = <math> \operatorname{Lognormal}\left( \mu,\,\sigma^2 \right) </math>
 | parameters = {{plainlist |
* <math> \mu \in ( -\infty, +\infty ) </math> (logarithm of [[location parameter|location]]),
* <math> \sigma > 0 </math> (logarithm of [[scale parameter|scale]])
}}
 | support = <math> x \in ( 0, +\infty ) </math>
 | pdf = <math> \frac{ 1 }{ x \sigma \sqrt{2\pi } } \exp\left( - \frac{ \left( \ln x - \mu \right)^2}{ 2 \sigma^2 } \right)</math>
 | cdf = <math>\begin{align}
&\frac{ 1 }{2}\left[1 + \operatorname{erf}\left( \frac{ \ln x - \mu }{\sigma\sqrt{2}} \right)\right] \\[1ex]
&= \Phi{\left(\frac{\ln x -\mu}{\sigma} \right)}
\end{align}</math>
 | quantile = <math>\begin{align}
&\exp\left( \mu + \sqrt{2\sigma^2}\operatorname{erf}^{-1}(2 p - 1) \right) \\[1ex]
& = \exp(\mu + \sigma \Phi^{-1}(p))
\end{align}</math>
 | mean = <math> \exp\left( \mu + \frac{\sigma^2}{2} \right) </math>
 | median = <math> \exp( \mu ) </math>
 | mode = <math> \exp\left( \mu - \sigma^2 \right) </math>
 | variance = <math> \left[ \exp(\sigma^2) - 1 \right] \exp\left( 2 \mu + \sigma^2\right ) </math>
 | skewness = <math> \left[ \exp\left( \sigma^2 \right) + 2 \right] \sqrt{\exp(\sigma^2) - 1 }</math>
 | kurtosis = <math> \exp\left( 4 \sigma^2 \right) + 2 \exp\left( 3 \sigma^2 \right) + 3 \exp\left( 2\sigma^2 \right) - 6 </math>
 | entropy = <math> \log_2 \left( \sqrt{2\pi e} \, \sigma e^{ \mu } \right) </math>
 | mgf = defined only for numbers with a {{nowrap|non-positive}} real part, see text
 | char = representation <math> \sum_{n=0}^{\infty} \frac{ {\left(i t\right)}^n }{ n! }e^{ n \mu + n^2 \sigma^2/2} </math> is asymptotically divergent, but adequate for most numerical purposes
 | fisher = <math> \frac{1}{\sigma^2} \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix} </math>
 | moments = <math> \mu = \ln \operatorname{E}[X] - \frac{1}{2} \ln\left( \frac{ \operatorname{Var}[X] }{ \operatorname{E}[X]^2 } + 1 \right),</math>
<br/>
<math> \sigma = \sqrt{ \ln \left( \frac{ \operatorname{Var}[X] }{ \operatorname{E}[X]^2 } + 1 \right) } </math>
 | ES = <math>\begin{align}
 &\frac{ e^{ \mu + \frac{ \sigma^2 }{2}} }{ 2p } \left[ 1 + \operatorname{erf} \left( \frac{ \sigma }{ \sqrt{2} } + \operatorname{erf}^{-1}(2p-1) \right) \right] \\[0.5ex]
 &= \frac{e^{ \mu + \frac{ \sigma^2 }{2}}}{1-p} \left[1 - \Phi(\Phi^{-1}(p) - \sigma)\right]
\end{align}</math><ref name="norton">{{cite journal | last1 = Norton | first1 = Matthew | last2 = Khokhlov | first2 = Valentyn | last3 = Uryasev | first3 = Stan | year = 2019 | title = Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation | journal = [[Annals of Operations Research]] | volume = 299 | issue = 1–2 | pages = 1281–1315 | publisher = Springer | doi = 10.1007/s10479-019-03373-1 | arxiv = 1811.11301 | s2cid = 254231768 | url = http://uryasev.ams.stonybrook.edu/wp-content/uploads/2019/10/Norton2019_CVaR_bPOE.pdf | archive-url = https://web.archive.org/web/20210418151110/http://uryasev.ams.stonybrook.edu/wp-content/uploads/2019/10/Norton2019_CVaR_bPOE.pdf | archive-date = 2021-04-18 | url-status = live | access-date = 2023-02-27 | via = stonybrook.edu }} </ref>
 }} <!-- end "infobox" -->

In [[probability theory]], a '''log-normal''' (or '''lognormal''') '''distribution''' is a continuous [[probability distribution]] of a [[random variable]] whose [[logarithm]] is [[normal distribution|normally distributed]]. Thus, if the random variable {{mvar|X}} is log-normally distributed, then {{math|1=''Y'' = ln ''X''}} has a normal distribution.<ref name=":1">{{Cite web | last = Weisstein | first = Eric W. | title = Log Normal Distribution | website = mathworld.wolfram.com | language = en | url = https://mathworld.wolfram.com/LogNormalDistribution.html | access-date = 2020-09-13 }}</ref><ref name=":2">{{cite web | title = 1.3.6.6.9. Lognormal Distribution | website = www.itl.nist.gov | publisher = U.S. [[National Institute of Standards and Technology]] (NIST) | url = https://www.itl.nist.gov/div898/handbook/eda/section3/eda3669.htm |access-date = 2020-09-13 }}</ref> Equivalently, if {{mvar|Y}} has a normal distribution, then the [[exponential function]] of {{mvar|Y}}, {{math|1=''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]].<ref name="JKB"/> The log-normal distribution has also been associated with other names, such as [[Donald MacAlister#log-normal|McAlister]], [[Gibrat's law|Gibrat]] and [[Cobb–Douglas]].<ref name="JKB"/>

A log-normal process is the statistical realization of the multiplicative [[mathematical product|product]] of many [[statistical independence|independent]] [[random variable]]s, 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 {{mvar|X}}—for which the mean and variance of {{math|ln ''X''}} are specified.<ref>{{cite journal | last1 = Park | first1 = Sung Y. | last2 = Bera | first2 = Anil K. | year = 2009 | title = Maximum entropy autoregressive conditional heteroskedasticity model | journal = Journal of Econometrics | volume = 150 | issue = 2 | pages = 219–230, esp. Table&nbsp;1, p.&nbsp;221 | citeseerx = 10.1.1.511.9750 | doi = 10.1016/j.jeconom.2008.12.014 | url = http://www.wise.xmu.edu.cn/Master/Download/..%5C..%5CUploadFiles%5Cpaper-masterdownload%5C2009519932327055475115776.pdf | url-status = dead | archive-url = https://web.archive.org/web/20160307144515/http://wise.xmu.edu.cn/uploadfiles/paper-masterdownload/2009519932327055475115776.pdf | archive-date = 2016-03-07 | access-date = 2011-06-02}}</ref>