Editing Log-normal distribution (section)

==Definitions==
===Generation and parameters===

Let <math> Z </math> be a [[Normal distribution#Standard normal distribution|standard normal variable]], and let <math>\mu</math> and <math>\sigma</math> be two real numbers, with {{nowrap|<math>\sigma > 0</math>.}} Then, the distribution of the random variable

<math display="block"> X = e^{\mu + \sigma Z} </math>

is called the log-normal distribution with parameters <math>\mu</math> and {{nowrap|<math>\sigma</math>.}} These are the [[expected value]] (or [[mean]]) and [[standard deviation]] of the variable's natural [[logarithm]], {{nowrap|<math>\ln X </math>,}} ''not'' the expectation and standard deviation of <math> X </math> itself.
[[File:Lognormal Distribution.svg|thumb|upright=1.5|Relation between normal and log-normal distribution. If <math> Y = \mu + \sigma Z </math> is normally distributed, then <math> X \sim e^Y </math> is log-normally distributed.]]

This relationship is true regardless of the base of the logarithmic or exponential function: If <math>\log_a X </math> is normally distributed, then so is <math>\log_b X </math> for any two positive numbers {{nowrap|<math> a, b \neq 1</math>.}} Likewise, if <math> e^Y </math> is log-normally distributed, then so is {{nowrap|<math> a^Y </math>,}} where {{nowrap|<math>0 < a \neq 1</math>.}}

In order to produce a distribution with desired mean <math>\mu_X</math> and variance {{nowrap|<math> \sigma_X^2</math>,}} one uses <math> \mu = \ln \frac{ \mu_X^2 }{ \sqrt{ \mu_X^2 + \sigma_X^2 } } </math> and {{nowrap|<math> \sigma^2 = \ln\left( 1 + \frac{ \sigma_X^2 }{ \mu_X^2 } \right) </math>.}}

Alternatively, the "multiplicative" or "geometric" parameters <math> \mu^* = e^\mu </math> and <math> \sigma^* = e^\sigma </math> can be used. They have a more direct interpretation: <math> \mu^* </math> is the '''''[[median]]''''' of the distribution, and <math> \sigma^* </math> is useful for determining "scatter" intervals, see below.

=== Probability density function ===

A positive random variable <math> X </math> is log-normally distributed (i.e., {{nowrap|<math display="inline"> X \sim \operatorname{Lognormal} \left( \mu, \sigma^2 \right) </math>),}} if the natural logarithm of <math> X </math> is normally distributed with mean <math> \mu</math> and variance {{nowrap|<math> \sigma^2</math>:}}

<math display="block"> \ln X \sim \mathcal{N}(\mu,\sigma^2)</math>

Let <math> \Phi </math> and <math> \varphi </math> be respectively the cumulative probability distribution function and the probability density function of the <math> \mathcal{N}( 0, 1 ) </math> standard normal distribution, then we have that<ref name=":1"/><ref name="JKB"/> the [[probability density function]] of the log-normal distribution is given by:

<math display="block">\begin{align}
f_X(x) & = \frac{d}{dx} \Pr\nolimits_X\left[ X \le x \right] \\[6pt]
& = \frac{d}{dx} \Pr\nolimits_X\left[ \ln X \le \ln x \right] \\[6pt]
& = \frac{d}{dx} \Phi{\left( \frac{ \ln x -\mu }{ \sigma } \right)} \\[6pt]
& = \varphi{\left( \frac{\ln x - \mu} \sigma \right)} \frac{d}{dx} \left( \frac{ \ln x - \mu }{ \sigma }\right) \\[6pt]
& = \varphi{\left( \frac{ \ln x - \mu }{ \sigma } \right)} \frac{ 1 }{ \sigma x } \\[6pt]
& = \frac{ 1 }{ x \sigma\sqrt{2 \pi } } \exp\left( -\frac{ (\ln x-\mu)^2 }{2 \sigma^2} \right) ~.
\end{align}</math>

=== Cumulative distribution function ===
<!-- erf changed into erfc because then the formula is slightly shorter, and besides the expression with erf is already present in the floating box -->
The [[cumulative distribution function]] is

<math display="block"> F_X(x) = \Phi{\left( \frac{\ln x - \mu} \sigma \right)} </math>

where <math> \Phi </math> is the cumulative distribution function of the standard normal distribution (i.e., {{nowrap|<math> \operatorname\mathcal{N}( 0, 1 ) </math>).}}

This may also be expressed as follows:<ref name=":1" />

<math display="block"> \frac{1}{2} \left[ 1 + \operatorname{erf} \left(\frac{\ln x - \mu}{\sigma\sqrt{2}}\right) \right] = \frac12 \operatorname{erfc} \left(-\frac{\ln x - \mu}{\sigma\sqrt{2}}\right) </math>

where {{big|{{math|erfc}}}} is the [[complementary error function]].

=== Multivariate log-normal ===
If <math>\boldsymbol X \sim \mathcal{N}(\boldsymbol\mu,\,\boldsymbol\Sigma)</math> is a [[multivariate normal distribution]], then <math>Y_i = \exp(X_i)</math> has a multivariate log-normal distribution.<ref>{{Cite conference | last = Tarmast | first = Ghasem | year = 2001 | title = Multivariate Log–Normal Distribution | url = http://isi.cbs.nl/iamamember/CD2/pdf/329.PDF | archive-url = https://web.archive.org/web/20130719214220/http://isi.cbs.nl/iamamember/CD2/pdf/329.PDF | archive-date = 2013-07-19 | url-status = live | conference = ISI Proceedings: 53rd Session | location = Seoul}}</ref><ref>{{Cite conference | last = Halliwell | first = Leigh | year = 2015 | title = The Lognormal Random Multivariate | url = http://www.casact.org/pubs/forum/15spforum/Halliwell.pdf | archive-url = https://web.archive.org/web/20150930111908/http://www.casact.org/pubs/forum/15spforum/Halliwell.pdf | archive-date = 2015-09-30 | url-status = live | conference = Casualty Actuarial Society E-Forum, Spring 2015 | location = Arlington, VA}}</ref> The exponential is applied element-wise to the random vector <math>\boldsymbol X</math>. The mean of <math>\boldsymbol Y</math> is

<math display="block">\operatorname{E}[\boldsymbol Y]_i = e^{\mu_i + \frac{1}{2} \Sigma_{ii}} ,</math>

and its [[covariance matrix]] is

<math display="block">\operatorname{Var}[\boldsymbol Y]_{ij} = e^{\mu_i + \mu_j + \frac{1}{2}(\Sigma_{ii} + \Sigma_{jj}) } \left( e^{\Sigma_{ij}} - 1\right) . </math>

Since the multivariate log-normal distribution is not widely used, the rest of this entry only deals with the [[univariate distribution]].

=== Characteristic function and moment generating function ===

All moments of the log-normal distribution exist and

<math display="block">\operatorname{E}[X^n] = e^{n\mu+n^2\sigma^2/2}</math>

This can be derived by letting <math display="inline">z = \tfrac{\ln x - \mu}{\sigma} - n \sigma</math> within the integral. However, the log-normal distribution is not determined by its moments.<ref name="Heyde">{{Citation | last = Heyde | first = CC. | title = On a Property of the Lognormal Distribution | work = Journal of the Royal Statistical Society, Series B | date = 2010 | volume = 25 | issue = 2 | pages = 392–393 | doi = 10.1007/978-1-4419-5823-5_6 | isbn = 978-1-4419-5822-8 | doi-access = free}}</ref> This implies that it cannot have a defined moment generating function in a neighborhood of zero.<ref>{{Cite book | last = Billingsley | first = Patrick | url = https://www.worldcat.org/oclc/780289503 | title = Probability and Measure | date = 2012 | publisher = Wiley | isbn = 978-1-118-12237-2 | edition = Anniversary | location = Hoboken, N.J. | pages = 415 | oclc = 780289503}}</ref> Indeed, the expected value <math>\operatorname{E}[e^{t X}]</math> is not defined for any positive value of the argument <math>t</math>, since the defining integral diverges.

The [[characteristic function (probability theory)|characteristic function]] <math>\operatorname{E}[e^{i t X}]</math> is defined for real values of {{mvar|t}}, but is not defined for any complex value of {{mvar|t}} that has a negative imaginary part, and hence the characteristic function is not [[Analytic function|analytic]] at the origin. Consequently, the characteristic function of the log-normal distribution cannot be represented as an infinite convergent series.<ref name="Holgate">{{Cite journal | last = Holgate | first = P. | year = 1989 | title = The lognormal characteristic function, vol. 18, pp. 4539–4548, 1989 | journal = Communications in Statistics – Theory and Methods | volume = 18 | issue = 12 | pages = 4539–4548 | doi = 10.1080/03610928908830173}}</ref> In particular, its Taylor [[formal series]] diverges:

<math display="block">\sum_{n=0}^\infty \frac{{\left(it\right)}^n}{n!} e^{n\mu + n^2\sigma^2/2}</math>

However, a number of alternative [[divergent series]] representations have been obtained.<ref name="Holgate" /><ref name="Barakat">{{Cite journal | last = Barakat | first = R. | year = 1976 | title = Sums of independent lognormally distributed random variables | journal = Journal of the Optical Society of America | volume = 66 | issue = 3 | pages = 211–216 | bibcode = 1976JOSA...66..211B | doi = 10.1364/JOSA.66.000211}}</ref><ref name="Barouch">{{Cite journal | last1 = Barouch | first1 = E. | last2 = Kaufman | first2 = GM. | last3 = Glasser | first3 = ML. | year = 1986 | title = On sums of lognormal random variables | url = http://dspace.mit.edu/bitstream/handle/1721.1/48703/onsumsoflognorma00baro.pdf | journal = Studies in Applied Mathematics | volume = 75 | issue = 1 | pages = 37–55 | doi = 10.1002/sapm198675137 | hdl = 1721.1/48703 | hdl-access = free }}</ref><ref name="Leipnik">{{Cite journal | last = Leipnik | first = Roy B. | date = January 1991 | title = On Lognormal Random Variables: I – The Characteristic Function | url = https://www.cambridge.org/core/services/aop-cambridge-core/content/view/F1563B5AD8918EF2CD51092F82EB0B73/S0334270000006901a.pdf/div-class-title-on-lognormal-random-variables-i-the-characteristic-function-div.pdf | journal = Journal of the Australian Mathematical Society, Series B | volume = 32 | issue = 3 | pages = 327–347 | doi = 10.1017/S0334270000006901 | doi-access = free }}</ref>

A closed-form formula for the characteristic function <math>\varphi(t)</math> with <math>t</math> in the domain of convergence is not known. A relatively simple approximating formula is available in closed form, and is given by<ref name="Asmussen">S. Asmussen, J.L. Jensen, L. Rojas-Nandayapa (2016). "On the Laplace transform of the Lognormal distribution",
[https://link.springer.com/article/10.1007/s11009-014-9430-7 Methodology and Computing in Applied Probability 18 (2), 441-458.]
[http://data.imf.au.dk/publications/thiele/2013/math-thiele-2013-06.pdf Thiele report 6 (13).]</ref>

<math display="block">\varphi(t) \approx \frac{\exp\left(-\frac{W^2(-it\sigma^2e^\mu) + 2W(-it\sigma^2e^\mu)}{2\sigma^2} \right)}{\sqrt{1 + W{\left(-it\sigma^2e^\mu\right)}}}</math>

where <math>W</math> is the [[Lambert W function]]. This approximation is derived via an asymptotic method, but it stays sharp all over the domain of convergence of <math>\varphi</math>.