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| {{Probability distribution
| | Hi there, I am Andrew Berryhill. As a woman what she truly likes is fashion and she's been performing it for quite a whilst. Office supervising is my profession. For a whilst I've been in Mississippi but now I'm considering other choices.<br><br>Also visit my web blog [http://xovibe.com/members/liamboyes/activity/314711/ psychic love readings] |
| | name = Log-normal
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| | type = continuous
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| | pdf_image = [[Image:Some log-normal distributions.svg|325px|Plot of the Lognormal PDF]]<br /><small>Some log-normal density functions with identical location parameter ''μ'' but differing scale parameters ''σ''</small>
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| | cdf_image = [[Image:Lognormal distribution CDF.svg|325px|Plot of the Lognormal CDF]]<br /><small>Cumulative distribution function of the log-normal distribution (with μ = 0 )</small>
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| | notation = <math>\ln\mathcal{N}(\mu,\,\sigma^2)</math>
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| | parameters= ''σ''<sup>2</sup> > 0 — shape (real), <br /> ''μ'' ∈ '''R''' — log-scale
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| | support = ''x'' ∈ (0, +∞)
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| | pdf = <math>\frac{1}{x\sqrt{2\pi}\sigma}\ e^{-\frac{\left(\ln x-\mu\right)^2}{2\sigma^2}}</math>
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| | cdf = <math>\frac12 + \frac12\,\mathrm{erf}\Big[\frac{\ln x-\mu}{\sqrt{2}\sigma}\Big]</math>
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| | mean = <math>e^{\mu+\sigma^2/2}</math>
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| | median = <math>e^{\mu}\,</math>
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| | mode = <math>e^{\mu-\sigma^2}</math>
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| | variance = <math>(e^{\sigma^2}\!\!-1) e^{2\mu+\sigma^2}</math>
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| | skewness = <math>(e^{\sigma^2}\!\!+2) \sqrt{e^{\sigma^2}\!\!-1}</math>
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| | kurtosis = <math>e^{4\sigma^2}\!\! + 2e^{3\sigma^2}\!\! + 3e^{2\sigma^2}\!\! - 6</math>
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| | entropy = <math>\frac12 + \frac12 \ln(2\pi\sigma^2) + \mu</math>
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| | mgf = (defined only on the negative half-axis, see text)
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| | char = representation <math>\sum_{n=0}^{\infty}\frac{(it)^n}{n!}e^{n\mu+n^2\sigma^2/2}</math> is asymptotically divergent but sufficient for numerical purposes
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| | fisher = <math>\begin{pmatrix}1/\sigma^2&0\\0&1/(2\sigma^4)\end{pmatrix}</math>
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| }}
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| In [[probability theory]], a '''log-normal distribution''' is a continuous [[probability distribution]] of a [[random variable]] whose [[logarithm]] is [[normal distribution|normally distributed]]. Thus, if the random variable <math>X</math> is log-normally distributed, then <math>Y = \log(X)</math> has a normal distribution. Likewise, if <math>Y</math> has a normal distribution, then <math>X = \exp(Y)</math> has a log-normal distribution. A random variable which is log-normally distributed takes only positive real values.
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| Log-normal is also written '''log normal''' or '''lognormal'''. The distribution is occasionally referred to as the '''Galton distribution''' or '''Galton's distribution''', after [[Francis Galton]].<ref name=JKB/> The log-normal distribution also has been associated with other names, such as McAlister, Gibrat and Cobb–Douglas.<ref name=JKB/>
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| A variable might be modeled as log-normal if it can be thought of as 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.) For example, in finance, the variable could represent the compound return from a sequence of many trades (each expressed as its return + 1); or a long-term [[Discounting#Discount factor|discount factor]] can be derived from the product of short-term discount factors. In wireless communication, the sas caused by shadowing or slow fading from random objects is often assumed to be log-normally distributed: see [[log-distance path loss model]].
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| The log-normal distribution is the [[maximum entropy probability distribution]] for a random variate ''X'' for which the mean and variance of <math>\ln(X)</math> are fixed.<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= |issue= |pages=219–230 |publisher=Elsevier |doi= |url=http://www.wise.xmu.edu.cn/Master/Download/..%5C..%5CUploadFiles%5Cpaper-masterdownload%5C2009519932327055475115776.pdf |accessdate=2011-06-02 }}</ref>
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| ==''μ'' and ''σ''==
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| In a log-normal distribution ''X'', the parameters denoted ''μ'' and ''σ'' are, respectively, the [[mean]] and [[standard deviation]] of the variable’s natural [[logarithm]] (by definition, the variable’s logarithm is normally distributed), which means
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| :<math> X=e^{\mu+\sigma Z} </math>
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| with ''Z'' a [[standard normal variable]].
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| This relationship is true regardless of the base of the logarithmic or exponential function. If log<sub>''a''</sub>(''Y'') is normally distributed, then so is log<sub>''b''</sub>(''Y''), for any two positive numbers ''a'', ''b'' ≠ 1. Likewise, if <math>e^X</math> is log-normally distributed, then so is <math>a^{X}</math>, where ''a'' is a positive number ≠ 1.
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| On a logarithmic scale, ''μ'' and ''σ'' can be called the ''location parameter'' and the ''scale parameter'', respectively.
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| In contrast, the mean and standard deviation of the non-logarithmized sample values are denoted ''m'' and ''s.d.'' in this article.
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| A log-normal distribution with mean ''m'' and variance ''v'' has parameters<ref>[http://www.mathworks.com/help/stats/lognstat.html "Lognormal mean and variance"]</ref>
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| :<math>
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| \mu=\ln\left(\frac{m^2}{\sqrt{v+m^2}}\right), \sigma=\sqrt{\ln\left(1+\frac{v}{m^2}\right)}
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| </math>
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| == Characterization ==
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| === Probability density function ===
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| The [[probability density function]] of a log-normal distribution is:<ref name=JKB/>
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| : <math>f_X(x;\mu,\sigma) = \frac{1}{x \sigma \sqrt{2 \pi}}\, e^{-\frac{(\ln x - \mu)^2}{2\sigma^2}},\ \ x>0</math>
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| This follows by applying the [[Probability density function#Dependent variables and change of variables|change-of-variables rule]] on the density function of a normal distribution.
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| === Cumulative distribution function ===
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| <!-- erf changed into erfc because then the formula is slightly shorter, and besides the expression with erf is already present in the floating box -->
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| The [[cumulative distribution function]] is
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| :<math>F_X(x;\mu,\sigma) = \frac12 \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) = \Phi\bigg(\frac{\ln x - \mu}{\sigma}\bigg),</math>
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| where erfc is the [[complementary error function]], and Φ is the cumulative distribution function of the [[standard normal]] distribution.
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| === Characteristic function and moment generating function ===
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| All moments of the log-normal distributions exist and it holds that
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| :<math>\operatorname{E}(X^n)=\mathrm{e}^{n\mu+\frac{n^2\sigma^2}{2}}</math>.
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| However, the moment generating function
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| :<math>\operatorname{E}(e^{t X})=\sum_{n=0}^\infty \frac{t^n}{n!}\operatorname{E}(X^n)</math>
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| does not converge.
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| The [[characteristic function (probability theory)|characteristic function]], E[''e''<sup> ''itX''</sup>], has a number of representations.{{Citation needed|date=February 2011}} The integral itself converges for [[Imaginary part|Im]](''t'') ≤ 0. The simplest representation is obtained by Taylor expanding ''e''<sup> ''itX''</sup> and using formula for moments below, giving{{Citation needed|date=February 2011}}
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| : <math>\varphi(t) = \sum_{n=0}^\infty \frac{(it)^n}{n!}e^{n\mu+n^2\sigma^2/2}.</math>
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| This series representation is divergent for [[Real part|Re]](''σ''<sup>2</sup>) > 0.{{Citation needed|date=February 2011}} However, it is sufficient for evaluating the characteristic function numerically at positive <math>\sigma </math> as long as the upper limit in the sum above is kept bounded, ''n'' ≤ ''N'', where
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| :<math>\max(|t|,|\mu|) \ll N \ll \frac{2}{\sigma^2}\ln\frac{2}{\sigma^2} </math>
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| and ''σ''<sup>2</sup> < 0.1.{{Citation needed|date=February 2011}} To bring the numerical values of parameters ''μ'', ''σ'' into the domain where strong inequality holds true one could use the fact that if ''X'' is log-normally distributed then ''X''<sup>''m''</sup> is also log-normally distributed with parameters ''μm'', ''σm''. Since <math> \mu\sigma^2 \propto m^3</math>, the inequality could be satisfied for sufficiently small ''m''. The sum of series first converges to the value of ''φ''(''t'') with arbitrary high accuracy if ''m'' is small enough, and left part of the strong inequality is satisfied. If considerably larger number of terms are taken into account the sum eventually diverges when the right part of the strong inequality is no longer valid.
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| Another useful representation is available<ref>Leipnik, Roy B. (1991), "On Lognormal Random Variables: I – The Characteristic Function", ''Journal of the Australian Mathematical Society Series B'', 32, 327–347.</ref><ref name="SUMS OF LOGNORMALS">Daniel Dufresne (2009), [http://www.soa.org/library/proceedings/arch/2009/arch-2009-iss1-dufresne.pdf, ''SUMS OF LOGNORMALS''], Centre for Actuarial Studies, University of Melbourne.</ref> by means of double Taylor expansion of ''e''<sup>(ln ''x'' − ''μ'')<sup>2</sup>/(2''σ''<sup>2</sup>)</sup>.
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| The [[moment-generating function]] for the log-normal distribution does not exist on the domain '''R''', but only exists on the half-interval <nowiki>(</nowiki>−∞, 0<nowiki>]</nowiki>.{{Citation needed|date=February 2011}}
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| == Properties ==
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| === Location and scale ===
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| For the log-normal distribution, the location and scale properties of the distribution are more readily treated using the [[geometric mean]] and [[geometric standard deviation]] than the [[arithmetic mean]] and standard deviation.
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| ====Geometric moments====
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| The [[geometric mean]] of the log-normal distribution is <math>e^{\mu}</math>. Because the log of a log-normal variable is symmetric and quantiles are preserved under monotonic transformations, the geometric mean of a log-normal distribution is equal to its median.<ref>Leslie E. Daly, Geoffrey Joseph Bourke (2000) [http://books.google.se/books?id=AY7LnYkiLNkC&pg=PA89 ''Interpretation and uses of medical statistics''] Edition: 5. Wiley-Blackwell ISBN 0-632-04763-1, ISBN 978-0-632-04763-5 (page 89)</ref>
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| The geometric mean (''m<sub>g</sub>'') can alternatively be derived from the arithmetic mean (''m<sub>a</sub>'') in a log-normal distribution by:
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| : <math> m_g = m_ae^{-\tfrac{1}{2}\sigma^2}.</math>
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| Note that the geometric mean is ''less'' than the arithmetic mean. This is due to the [[AM–GM inequality]], and corresponds to the logarithm being convex down. The correction term <math>e^{-\tfrac{1}{2}\sigma^2}</math> can accordingly be interpreted as a [[convexity correction]]. From the point of view of [[stochastic calculus]], this is the same correction term as in [[Itō's lemma#Geometric Brownian motion|Itō's lemma for geometric Brownian motion]].
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| The [[geometric standard deviation]] is equal to <math>e^{\sigma}</math>.{{Citation needed|date=February 2011}}
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| ====Arithmetic moments====
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| If ''X'' is a lognormally distributed variable, its [[expected value]] (E – the [[arithmetic mean]]), [[variance]] (Var), and [[standard deviation]] (s.d.) are
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| : <math>\begin{align}
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| & \operatorname{E}[X] = e^{\mu + \tfrac{1}{2}\sigma^2}, \\
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| & \operatorname{Var}[X] = (e^{\sigma^2} - 1) e^{2\mu + \sigma^2} = (e^{\sigma^2} - 1)(\operatorname{E}[X])^2\\
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| & \operatorname{s.d.}[X] = \sqrt{\operatorname{Var}[X]} = e^{\mu + \tfrac{1}{2}\sigma^2}\sqrt{e^{\sigma^2} - 1}.
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| \end{align}</math>
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| Equivalently, parameters ''μ'' and ''σ'' can be obtained if the expected value and variance are known; it is simpler if ''σ'' is computed first:
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| : <math>\begin{align}
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| \mu &= \ln(\operatorname{E}[X]) - \frac12 \ln\!\left(1 + \frac{\mathrm{Var}[X]}{(\operatorname{E}[X])^2}\right) = \ln(\operatorname{E}[X]) - \frac12 \sigma^2, \\
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| \sigma^2 &= \ln\!\left(1 + \frac{\operatorname{Var}[X]}{(\operatorname{E}[X])^2}\right).
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| \end{align}</math>
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| For any real or complex number ''s'', the ''s''<sup>th</sup> [[moment (mathematics)|moment]] of log-normal ''X'' is given by<ref name=JKB/>
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| : <math>\operatorname{E}[X^s] = e^{s\mu + \tfrac{1}{2}s^2\sigma^2}.</math>
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| A log-normal distribution is not uniquely determined by its moments E[''X''<sup>''k''</sup>] for ''k'' ≥ 1, that is, there exists some other distribution with the same moments for all ''k''.<ref name=JKB/> In fact, there is a whole family of distributions with the same moments as the log-normal distribution.{{Citation needed|date=March 2012}}
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| === Mode and median ===
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| [[Image:Comparison mean median mode.svg|thumb|300px|Comparison of [[mean]], [[median]] and [[mode (statistics)|mode]] of two log-normal distributions with different [[skewness]].]]
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| The [[mode (statistics)|mode]] is the point of global maximum of the probability density function. In particular, it solves the equation (ln ''ƒ'')′ = 0:
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| : <math>\mathrm{Mode}[X] = e^{\mu - \sigma^2}.</math>
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| The [[median]] is such a point where ''F<sub>X</sub>'' = 1/2:
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| :<math>\mathrm{Med}[X] = e^\mu\,.</math>
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| === Coefficient of variation ===
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| The [[coefficient of variation]] is the ratio ''s.d.'' over ''m'' (on the natural scale)
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| and is equal to:
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| : <math>\sqrt{e^{\sigma^2}\!\!-1}</math>
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| === Partial expectation ===
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| The partial expectation of a random variable ''X'' with respect to a threshold ''k'' is defined as <math> g(k) = \int_k^\infty \!xf(x)\, dx </math> where <math> f(x)</math> is the probability density function of ''X''. Alternatively, and using the definition of [[conditional expectation]], it can be written as ''g''(''k'')=E[''X'' | ''X'' > ''k'']*P(''X'' > ''k''). For a log-normal random variable the partial expectation is given by:
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| :<math>g(k) = \int_k^\infty \!xf(x)\, dx
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| = e^{\mu+\tfrac{1}{2}\sigma^2}\, \Phi\!\left(\frac{\mu+\sigma^2-\ln k}{\sigma}\right).</math>
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| The derivation of the formula is provided in the discussion of this Wikipedia entry. The partial expectation formula has applications in insurance and economics, it is used in solving the partial differential equation leading to the [[Black–Scholes formula]].
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| === Other ===
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| A set of data that arises from the log-normal distribution has a symmetric [[Lorenz curve]] (see also [[Lorenz asymmetry coefficient]]).<ref name=EcolgyArticle>{{cite journal
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| | doi = 10.1890/0012-9658(2000)081[1139:DIIPSO]2.0.CO;2
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| | last1 = Damgaard
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| | first1 = Christian
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| | first2 = Jacob |last2=Weiner
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| | title = Describing inequality in plant size or fecundity
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| | journal = Ecology
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| | year = 2000 | volume = 81 | issue = 4 | pages = 1139–1142
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| }}</ref>
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| The harmonic (''H''), geometric (''G'') and arithmetic (''A'') means of this distribution are related;<ref name=Rossman1990>Rossman LA (1990) "Design stream flows based on harmonic means". ''J Hydraulic Engineering ASCE'' 116 (7) 946–950</ref> such relation is given by
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| : <math>H = \frac{G^2}{ A} .</math>
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| Log-normal distributions are [[infinite divisibility (probability)|infinitely divisible]].<ref name=JKB/>
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| ==Occurrence==
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| The log-normal distribution is important in the description of natural phenomena.
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| The reason is that for many natural processes of growth, [[growth rate]] is independent of size. This is also known as [[Gibrat's law]], after Robert Gibrat (1904–1980) who formulated it for firms (companies). It can be shown that a growth process following Gibrat's law will result in entity sizes with a log-normal distribution.<ref>Sutton, J. (1997), "Gibrat's Legacy", Journal of Economic Literature XXXV, 40–59.</ref> Examples include:
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| * In [[biology]] and [[medicine]],
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| ** Measures of size of living tissue (length, skin area, weight);<ref>{{cite book
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| | last = Huxley | first = Julian S.
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| | year = 1932
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| | title = Problems of relative growth
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| | publisher = London
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| | oclc = 476909537
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| | ref = harv
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| | isbn = 0-486-61114-0
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| }}</ref>
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| ** For highly communicable epidemics, such as SARS in 2003, if publication intervention is involved, the number of hospitalized cases is shown to satistfy the lognormal distribution with no free parameters if an entropy is assumed and the standard deviation is determined by the principle of maximum rate of entropy production.<ref name=Wang />
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| ** The ''length'' of ''inert'' appendages (hair, claws, nails, teeth) of biological specimens, ''in the direction of growth'';{{Citation needed|date=February 2011}}
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| ** Certain physiological measurements, such as blood pressure of adult humans (after separation on male/female subpopulations)<ref>{{cite journal|last=Makuch|first=Robert W.|coauthors=D.H. Freeman, M.F. Johnson|title=Justification for the lognormal distribution as a model for blood pressure|journal=Journal of Chronic Diseases|year=1979|volume=32|issue=3|pages=245–250|doi=10.1016/0021-9681(79)90070-5. (http://www.sciencedirect.com/science/article/pii/0021968179900705|url=http://www.sciencedirect.com/science/article/pii/0021968179900705|accessdate=27 February 2012}}</ref>
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| [[File:FitLogNormDistr.tif|thumb|200px|Fitted cumulative log-normal distribution to annually maximum 1-day rainfalls, see [[distribution fitting]] ]]
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| :Consequently, [[reference ranges]] for measurements in healthy individuals are more accurately estimated by assuming a log-normal distribution than by assuming a symmetric distribution about the mean.
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| * In [[colloidal chemistry]],
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| ** Particle size distributions
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| *In [[hydrology]], the log-normal distribution is used to analyze extreme values of such variables as monthly and annual maximum values of daily rainfall and river discharge volumes.<ref>{{cite book|last=Ritzema (ed.)|first=H.P.|title=Frequency and Regression Analysis|year=1994|publisher=Chapter 6 in: Drainage Principles and Applications, Publication 16, International Institute for Land Reclamation and Improvement (ILRI), Wageningen, The Netherlands|pages=175–224|url=http://www.waterlog.info/pdf/freqtxt.pdf|isbn=90-70754-33-9}}</ref>
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| **The image on the right illustrates an example of fitting the log-normal distribution to ranked annually maximum one-day rainfalls showing also the 90% [[confidence belt]] based on the [[binomial distribution]]. The rainfall data are represented by [[plotting position]]s as part of a [[cumulative frequency analysis]].
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| *in social sciences and demographics
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| ** In [[economics]], there is evidence that the [[income]] of 97%–99% of the population is distributed log-normally.<ref>Clementi, Fabio; [[Mauro Gallegati|Gallegati, Mauro]] (2005) [http://ideas.repec.org/p/wpa/wuwpmi/0505006.html "Pareto's law of income distribution: Evidence for Germany, the United Kingdom, and the United States"], EconWPA</ref>
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| ** In [[finance]], in particular the [[Black–Scholes model]], changes in the ''logarithm'' of exchange rates, price indices, and stock market indices are assumed normal<ref>Black, Fischer and Myron Scholes, "The Pricing of Options and Corporate Liabilities", Journal of Political Economy, Vol. 81, No. 3, (May/June 1973), pp. 637–654.</ref> (these variables behave like compound interest, not like simple interest, and so are multiplicative). However, some mathematicians such as [[Benoît Mandelbrot]] have argued <ref>{{cite book|last=Madelbrot|first=Beniot|title=The (mis-)Behaviour of Markets|year=2004|url=http://books.google.com.au/books/about/The_mis_behavior_of_markets.html?id=9w15j-Ka0vgC&redir_esc=y}}</ref> that [[Lévy skew alpha-stable distribution|log-Lévy distributions]] which possesses [[heavy tails]] would be a more appropriate model, in particular for the analysis for [[stock market crash]]es. Indeed stock price distributions typically exhibit a [[fat tail]].<ref>Bunchen, P., ''Advanced Option Pricing'', University of Sydney coursebook, 2007</ref>
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| ** [[Historical urban community sizes|city sizes]]
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| *technology
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| ** In [[Reliability (statistics)|reliability]] analysis, the lognormal distribution is often used to model times to repair a maintainable system.<ref>{{cite book
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| | last = O'Connor | first = Patrick
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| | last2 = Kleyner | first2 = Andre
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| | year = 2011
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| | title = Practical Reliability Engineering
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| | publisher = John Wiley & Sons
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| | isbn = 978-0-470-97982-2
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| | page = 35
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| }}</ref>
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| ** In [[wireless communication]], "the local-mean power expressed in logarithmic values, such as dB or neper, has a normal (i.e., Gaussian) distribution." <ref>http://wireless.per.nl/reference/chaptr03/shadow/shadow.htm {{dead link|date=July 2012}}</ref>
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| ** It has been proposed that coefficients of friction and wear may be treated as having a lognormal distribution <ref>{{cite doi|10.1016/j.ress.2007.09.005}}</ref>
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| ** In spray process, such as droplet impact, the size of secondary produced droplet has a lognormal distribution, with the standard deviation ：<math>\sigma=\frac{\sqrt{6}}{6}</math> determined by the principle of maximum rate of entropy production<ref name=wu>{{cite journal|last=Wu|first=Zi-Niu|title=Prediction of the size distribution of secondary ejected droplets by crown splashing of droplets impinging on a solid wall|journal=Probabilistic Engineering Mechanics|date=July 2003|volume=18|issue=3|pages=241–249|doi=doi.org/10.1016/S0266-8920(03)00028-6}}</ref> If the lognormal distribution is inserted into the Shannon entropy expression and if the rate of entropy production is maximized (principle of maximum rate of entropy production), then σ is given by :<math>\sigma=\frac{1}{\sqrt{6}}</math><ref name=wu /> and with this parameter the droplet size distribution for spray process is well predicted. It is an open question whether this value of σ has some generality for other cases, though for spreading of communicable epidemics, σ is shown also to take this value.<ref name=Wang>{{cite web|last=WB|first=Wang|coauthors=CF Wang, ZN Wu and RF Hu |year=2013|title=Modelling the spreading rate of controlled communicable epidemics through an entropy-based thermodynamic model|url=http://phys.scichina.com:8083/sciGe/EN/10.1007/s11433-013-5321-0|publisher=SCIENCE CHINA Physics, Mechanics & Astronomy|volume=56 |issue=11|pages=2143-2150}}</ref>
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| == Maximum likelihood estimation of parameters ==
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| For determining the [[maximum likelihood]] estimators of the log-normal distribution parameters μ and σ, we can use the [[normal distribution#Estimation of parameters|same procedure]] as for the [[normal distribution]]. To avoid repetition, we observe that
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| : <math>f_L (x;\mu, \sigma) = \prod_{i=1}^n \left(\frac 1 x_i\right) \, f_N (\ln x; \mu, \sigma)</math>
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| where by ''ƒ''<sub>''L''</sub> we denote the probability density function of the log-normal distribution and by ''ƒ''<sub>''N''</sub> that of the normal distribution. Therefore, using the same indices to denote distributions, we can write the log-likelihood function thus:
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| : <math>
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| \begin{align}
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| \ell_L (\mu,\sigma | x_1, x_2, \dots, x_n)
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| & {} = - \sum _k \ln x_k + \ell_N (\mu, \sigma | \ln x_1, \ln x_2, \dots, \ln x_n) \\
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| & {} = \operatorname {constant} + \ell_N (\mu, \sigma | \ln x_1, \ln x_2, \dots, \ln x_n).
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| \end{align}
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| </math>
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| Since the first term is constant with regard to μ and σ, both logarithmic likelihood functions, ''ℓ''<sub>''L''</sub> and ''ℓ''<sub>''N''</sub>, reach their maximum with the same ''μ'' and ''σ''. Hence, using the formulas for the normal distribution maximum likelihood parameter estimators and the equality above, we deduce that for the log-normal distribution it holds that
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| : <math>\widehat \mu = \frac {\sum_k \ln x_k} n,
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| \widehat \sigma^2 = \frac {\sum_k \left( \ln x_k - \widehat \mu \right)^2} {n}.</math>
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| == Multivariate log-normal ==
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| If <math>\boldsymbol X \sim \mathcal{N}(\boldsymbol\mu,\,\boldsymbol\Sigma)</math> is a [[multivariate normal distribution]] then <math>\boldsymbol Y=\exp(\boldsymbol X)</math> has a multivariate log-normal distribution<ref>Tarmast, Ghasem (2001) [http://isi.cbs.nl/iamamember/CD2/pdf/329.PDF "Multivariate Log–Normal Distribution"] ''ISI Proceedings: Seoul 53rd Session 2001''</ref> with mean
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| :<math>\operatorname{E}[\boldsymbol Y]_i=e^{\mu_i+\frac{1}{2}\Sigma_{ii}} ,</math>
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| and [[covariance matrix]]
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| :<math>\operatorname{Var}[\boldsymbol Y]_{ij}=e^{\mu_i+\mu_j + \frac{1}{2}(\Sigma_{ii}+\Sigma_{jj}) }( e^{\Sigma_{ij}} - 1) . </math>
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| == Generating log-normally distributed random variates ==
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| Given a random variate ''Z'' drawn from the [[normal distribution]] with 0 mean and 1 standard deviation, then the variate
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| :<math>X= e^{\mu + \sigma Z}\,</math>
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| has a log-normal distribution with parameters ''<math>\mu</math>'' and <math>\sigma</math>.
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| == Related distributions ==
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| * If <math>X \sim \mathcal{N}(\mu, \sigma^2)</math> is a [[normal distribution]], then <math>\exp(X) \sim \operatorname{Log-\mathcal{N}}(\mu, \sigma^2).</math>
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| * If <math>X \sim \operatorname{Log-\mathcal{N}}(\mu, \sigma^2)</math> is distributed log-normally, then <math>\ln(X) \sim \mathcal{N}(\mu, \sigma^2)</math> is a normal random variable.
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| * If <math>X_j \sim \operatorname{Log-\mathcal{N}}(\mu_j, \sigma_j^2)</math> are ''n'' [[statistical independence|independent]] log-normally distributed variables, and <math>Y = \textstyle\prod_{j=1}^n X_j</math>, then ''Y'' is also distributed log-normally:
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| : <math>Y \sim \operatorname{Log-\mathcal{N}}\Big(\textstyle \sum_{j=1}^n\mu_j,\ \sum_{j=1}^n \sigma_j^2 \Big).</math>
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| * Let <math>X_j \sim \operatorname{Log-\mathcal{N}}(\mu_j,\sigma_j^2)\ </math> be independent log-normally distributed variables with possibly varying ''σ'' and ''μ'' parameters, and <math>Y=\textstyle\sum_{j=1}^n X_j</math>. The distribution of ''Y'' has no closed-form expression, but can be reasonably approximated by another log-normal distribution ''Z'' at the right tail. Its probability density function at the neighborhood of 0 has been characterized<ref name=Gao/> and it does not resemble any log-normal distribution. A commonly used approximation (due to L.F. Fenton, but previously stated by R.I. Wilkinson without mathematical justification<ref name="SUMS OF LOGNORMALS"/>) is obtained by matching the mean and variance:
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| : <math>\begin{align}
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| \sigma^2_Z &= \log\!\left[ \frac{\sum e^{2\mu_j+\sigma_j^2}(e^{\sigma_j^2}-1)}{(\sum e^{\mu_j+\sigma_j^2/2})^2} + 1\right], \\
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| \mu_Z &= \log\!\left[ \sum e^{\mu_j+\sigma_j^2/2} \right] - \frac{\sigma^2_Z}{2}.
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| \end{align}</math>
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| In the case that all <math>X_j</math> have the same variance parameter <math>\sigma_j=\sigma</math>, these formulas simplify to
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| : <math>\begin{align}
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| \sigma^2_Z &= \log\!\left[ (e^{\sigma^2}-1)\frac{\sum e^{2\mu_j}}{(\sum e^{\mu_j})^2} + 1\right], \\
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| \mu_Z &= \log\!\left[ \sum e^{\mu_j} \right] + \frac{\sigma^2}{2} - \frac{\sigma^2_Z}{2}.
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| \end{align}</math>
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| * If <math>X \sim \operatorname{Log-\mathcal{N}}(\mu, \sigma^2)</math>, then ''X'' + ''c'' is said to have a ''shifted log-normal'' distribution with support ''x'' ∈ (c, +∞). E[''X'' + ''c''] = E[''X''] + ''c'', Var[''X'' + ''c''] = Var[''X''].
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| * If <math>X \sim \operatorname{Log-\mathcal{N}}(\mu, \sigma^2)</math>, then <math>a X \sim \operatorname{Log-\mathcal{N}}( \mu + \ln a,\ \sigma^2).</math>
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| * If <math>X \sim \operatorname{Log-\mathcal{N}}(\mu, \sigma^2)</math>, then <math>\tfrac{1}{X} \sim \operatorname{Log-\mathcal{N}}(-\mu,\ \sigma^2).</math>
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| * If <math>X \sim \operatorname{Log-\mathcal{N}}(\mu, \sigma^2)</math> then <math>X^a \sim \operatorname{Log-\mathcal{N}}(a\mu,\ a^2 \sigma^2).</math> for <math>a \neq 0\, </math>
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| * Lognormal distribution is a special case of semi-bounded [[Johnson distribution]]
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| * If <math>X|Y \sim \mathrm{Rayleigh}(Y)\,</math> with <math> Y \sim \operatorname{Log-\mathcal{N}}(\mu, \sigma^2)</math>, then <math> X \sim \mathrm{Suzuki}(\mu, \sigma)\,</math> ([[Suzuki distribution]])
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| == Similar distributions ==
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| A substitute for the log-normal whose integral can be expressed in terms of more elementary functions<ref>{{cite DOI|10.1061/(ASCE)1084-0699(2002)7:6(441)}}</ref> can be obtained based on the [[logistic distribution]] to get an approximation for the [[Cumulative distribution function|CDF]]
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| :<math> F(x;\mu,\sigma) = \left[\left(\frac{e^\mu}{x}\right)^{\pi/(\sigma \sqrt{3})} +1\right]^{-1}.</math>
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| This is a [[log-logistic distribution]].
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| == See also ==
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| * [[Log-distance path loss model]]
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| * [[Slow fading]]
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| * [[Stochastic volatility]]
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| == Notes ==
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| {{Reflist|refs=
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| <ref name=JKB>{{Citation | last1=Johnson | first1=Norman L. | last2=Kotz | first2=Samuel | last3=Balakrishnan | first3=N. | title=Continuous univariate distributions. Vol. 1 | publisher=[[John Wiley & Sons]] | location=New York | edition=2nd | series=Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics | isbn=978-0-471-58495-7 | mr = 1299979| year=1994 |chapter=14: Lognormal Distributions}}</ref>
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| <ref name=Gao>Gao, X.; Xu, H; Ye, D. (2009), [http://www.hindawi.com/journals/ijmms/2009/630857.html "Asymptotic Behaviors of Tail Density for Sum of Correlated Lognormal Variables"]. International Journal of Mathematics and Mathematical Sciences, vol. 2009, Article ID 630857. {{doi|10.1155/2009/630857}}</ref>
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| <ref name=wu>Wu, Z.N. (2003), [ "Prediction of the size distribution of secondary ejected droplets by crown splashing of droplets impinging on a solid wall"]. Probabilistic Engineering Mechanics, Volume 18, Issue 3, July 2003, Pages 241–249. {{doi|10.1016/S0266-8920(03)00028-6}}</ref>
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| }}
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| ==References==
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| * Aitchison, J. and Brown, J.A.C. (1957) ''The Lognormal Distribution'', Cambridge University Press.
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| * E. Limpert, W. Stahel and M. Abbt (2001) ''[http://stat.ethz.ch/~stahel/lognormal/bioscience.pdf Log-normal Distributions across the Sciences: Keys and Clues]'', BioScience, 51 (5), 341–352.
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| * [[Eric W. Weisstein]] et al. [http://mathworld.wolfram.com/LogNormalDistribution.html Log Normal Distribution] at [[MathWorld]]. Electronic document, retrieved October 26, 2006.
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| * {{cite DOI|10.1080/03610928908830173}}
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| == Further reading ==
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| * Robert Brooks, Jon Corson, and [[Jimbo Wales|J. Donal Wales.]] [http://papers.ssrn.com/sol3/papers.cfm?abstract_id=5735 "The Pricing of Index Options When the Underlying Assets All Follow a Lognormal Diffusion"], in ''Advances in Futures and Options Research'', volume 7, 1994.
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| == External links ==
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| {{commons category}}
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| {{ProbDistributions|continuous-semi-infinite}}
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| {{Common univariate probability distributions}}
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| {{DEFAULTSORT:Log-Normal Distribution}}
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| [[Category:Continuous distributions]]
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| [[Category:Normal distribution]]
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| [[Category:Exponential family distributions]]
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| [[Category:Probability distributions]]
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