Log-normal distribution

Probability distribution
name =Log-normal
type =density
pdf_

μ=0
cdf_

μ=0
parameters =$sigma\; >\; 0$

support =$[0,+infty)!$
pdf =$frac\{1\}\{xsigmasqrt\{2pi$expleft [-frac{left(ln(x)-mu ight)^2}{2sigma^2} ight]
cdf =$frac\{1\}\{2\}+frac\{1\}\{2\}\; mathrm\{erf\}left\; [frac\{ln(x)-mu\}\{sigmasqrt\{2\; ight]$
mean =$e^\{mu+sigma^2/2\}$
median =$e^\{mu\},$
mode =$e^\{mu-sigma^2\}$
variance =$(e^\{sigma^2\}!!-1)\; e^\{2mu+sigma^2\}$
skewness =$(e^\{sigma^2\}!!+2)sqrt\{e^\{sigma^2\}!!-1\}$
kurtosis =$\{e^\{4sigma^2\}+2e^\{3sigma^2\}+3e^\{2sigma^2\}-6\}$
entropy =$frac\{1\}\{2\}+frac\{1\}\{2\}ln(2pisigma^2)\; +\; mu$
mgf =(see text for raw moments)
char =

In probability and statistics, the log-normal distribution is the single-tailed probability distribution of any random variable whose logarithm is normally distributed. If "X" is a random variable with a normal distribution, then "Y" = exp("X") has a log-normal distribution; likewise, if "Y" is log-normally distributed, then log("Y") is normally distributed. (The base of the logarithmic function does not matter: if log_"a"("Y") is normally distributed, then so is log_"b"("Y"), for any two positive numbers "a", "b" ≠ 1.)

Log-normal is also written log normal or lognormal.

A variable might be modeled as log-normal if it can be thought of as the multiplicative product of many independent factors which are positive and close to 1. For example the long-term return rate on a stock investment can be considered to be the product of the daily return rates. In wireless communication, the attenuation caused by shadowing or slow fading from random objects is often assumed to be log-normally distributed. See log-distance path loss model‎.

Characterization

Probability density function

The log-normal distribution has the probability density function

: $f(x;mu,sigma)\; =\; frac\{1\}\{x\; sigma\; sqrt\{2\; pie^\{-frac\{(ln\; (x)\; -\; mu)^2\}\{2sigma^2$

for "x" > 0, where "μ" and "σ" are the mean and standard deviation of the variable's natural logarithm (by definition, the variable's logarithm is normally distributed). These parameters are in this context measured in neper, provided that natural logarithms are used, but is in the context of wireless communication typically measured in decibel.

Cumulative distribution function

:$frac\{1\}\{2\}+frac\{1\}\{2\}\; mathrm\{erf\}left\; [frac\{ln(x)-mu\}\{sigmasqrt\{2\; ight]$

Properties

Mean and standard deviation

The expected value (mean) is

:$mathrm\{E\}(X)\; =\; e^\{mu\; +\; sigma^2/2\},!$

and the variance is

:$mathrm\{Var\}(X)\; =\; (e^\{sigma^2\}\; -\; 1)\; e^\{2mu\; +\; sigma^2\},!$

hence the standard deviation is

:$mathrm\{Std\; Dev\}(X)\; =\; sqrt\{mathrm\{Var\}(X)\}\; =\; sqrt\{(e^\{sigma^2\}\; -\; 1)\}\; e^\{mu\; +\; sigma^2/2\},!$

Equivalent relationships may be written to obtain $mu,!$ and $sigma,!$ given the expected value and variance:

:$mu\; =\; ln(mathrm\{E\}(X))-frac\{1\}\{2\}lnleft(1+frac\{mathrm\{Var\}(X)\}\{(mathrm\{E\}(X))^2\}\; ight),!$

:$sigma^2\; =\; lnleft(frac\{mathrm\{Var\}(X)\}\{(mathrm\{E\}(X))^2\}+1\; ight),!$

Mode and median

The mode is

:$mathrm\{Mode\}(X)\; =\; e^\{mu\; -\; sigma^2\},!$

The median is

:$ilde\{X\}\; =\; e^\{mu\},!$

Geometric mean and geometric standard deviation

The geometric mean of the log-normal distribution is $e^\{mu\},!$, and the geometric standard deviation is equal to $e^\{sigma\},!$.

If a sample of data is determined to come from a log-normally distributed population, the geometric mean and the geometric standard deviation may be used to estimate confidence intervals akin to the way the arithmetic mean and standard deviation are used to estimate confidence intervals for a normally distributed sample of data.

Where geometric mean $mu\_mathrm\{geo\}\; =\; exp(mu),!$ and geometric standard deviation $sigma\_mathrm\{geo\}\; =\; exp(sigma),!$

Moments

All moments exist and are given by:$operatorname\{E\}(X^s)=e^\{smu+s^2sigma^2/2\}$for any real number "s". A log-normal distribution is not uniquely determined by its moments $operatorname\{E\}(X^k)$ for $kge1$, that is, there exists some other distribution with the same moments for all "k".

Moment generating function

The moment-generating function for the log-normal distribution does not exist.

Partial expectation

The partial expectation of a random variable "X" with respect to a threshold "k" is defined as

:$g(k)=int\_k^infty\; x\; f(x),\; dx,!$

where $f(x),!$ is the density. For a lognormal density it can be shown that

:$g(k)=exp(mu+sigma^2/2)Phileft(frac\{-ln(k)+mu+sigma^2\}\{sigma\}\; ight),!$

where $scriptstylePhi,!$ is the cumulative distribution function of the standard normal. The partial expectation of a lognormal has applications in insurance and in economics (for example it can be used to derive the Black–Scholes formula).

Maximum likelihood estimation of parameters

For determining the maximum likelihood estimators of the log-normal distribution parameters μ and σ, we can use the same procedure as for the normal distribution. To avoid repetition, we observe that

: $f\_L\; (x;mu,\; sigma)\; =\; frac\; 1\; x\; ,\; f\_N\; (ln\; x;\; mu,\; sigma)$

where by "ƒ"_"L" we denote the probability density function of the log-normal distribution and by "ƒ"_"N" that of the normal distribution. Therefore, using the same indices to denote distributions, we can write the log-likelihood function thus:

:

Since the first term is constant with regard to μ and σ, both logarithmic likelihood functions, $ell\_L$ and $ell\_N$, 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

: $widehat\; mu\; =\; frac\; \{sum\_k\; ln\; x\_k\}\; n,\; widehat\; sigma^2\; =\; frac\; \{sum\_k\; \{left(\; ln\; x\_k\; -\; widehat\; mu\; ight)^2\; \{n\}.$

Related distributions

* If $X\; sim\; N(mu,\; sigma^2)$ is a normal distribution then $exp(X)\; sim\; operatorname\{Log-N\}(mu,\; sigma^2)$.
* If $X\_m\; sim\; operatorname\; \{Log-N\}\; (mu,\; sigma\_m^2),\; m\; =\; 1,dots,\; n$ are independent log-normally distributed variables with the same "μ" parameter and possibly varying "σ", and $Y\; =\; prod\_\{m=1\}^n\; X\_m$, then "Y" is a log-normally distributed variable as well::: $Y\; sim\; operatorname\; \{Log-N\}\; left(\; nmu,\; sum\; \_\{m=1\}^n\; sigma\_m^2\; ight).$
* Let $X\_m\; sim\; operatorname\; \{Log-N\}\; (mu\_m,sigma\_m^2),\; m=\{1,dots,n\}$ be independent log-normally distributed variables with possibly varying "σ" and "μ" parameters, and$Y=sum\_\{m=1\}^n\; X\_m$. The distribution of "Y" has no closed-formexpression, but can be reasonably approximated by another log-normal distribution "Z". A commonly used approximation (due to Fenton and Wilkinson) is obtained by matching the mean and variance:::$sigma^2\_Z\; =\; logleft\; [\; frac\{sum\; e^\{2mu\_m+sigma\_m^2\}(e^\{sigma\_m^2\}-1)\}\{(sum\; e^\{mu\_m+sigma\_m^2/2\})^2\}+1\; ight]$::$mu\_Z\; =\; logleft(\; sum\; e^\{mu\_m+sigma\_m^2/2\}\; ight)-\; frac\{sigma^2\_Z\}\{2\}.$In the case that all $X\_m$ have the same variance parameter $sigma\_m=sigma$, these formulas simplify to::$sigma^2\_Z\; =\; logleft\; [\; (e^\{sigma^2\}-1)frac\{sum\; e^\{2mu\_m\{(sum\; e^\{mu\_m\})^2\}+1\; ight]$::$mu\_Z\; =\; logleft(\; sum\; e^\{mu\_m\}\; ight)\; +\; frac\{sigma^2\}\{2\}\; -\; frac\{sigma^2\_Z\}\{2\}.$
* A substitute for the log-normal whose integral can be expressed in terms of more elementary functions (Swamee, 2002) can be obtained based on the logistic distribution to get the CDF::$F(x;mu,sigma)\; =\; left\; [left(frac\{e^mu\}\{x\}\; ight)^\{pi/(sigma\; sqrt\{3\})\}\; +1\; ight]\; ^\{-1\}.$This is a log-logistic distribution.
* If $X\; sim\; operatorname\; \{Log-N\}\; (mu,\; sigma^2)$ then $X\; +\; c$ is called shifted log-normal.

Further reading

*Robert Brooks, Jon Corson, and 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.

References

*"The Lognormal Distribution", Aitchison, J. and Brown, J.A.C. (1957)
*" [http://stat.ethz.ch/~stahel/lognormal/bioscience.pdf Log-normal Distributions across the Sciences: Keys and Clues] ", E. Limpert, W. Stahel and M. Abbt,. BioScience, 51 (5), p. 341–352 (2001).
*" [http://www.stat.nctu.edu.tw/subhtml/E_source/teachers_eng/jclee/course/chapter5.doc Normal and Lognormal Distribution] ", in Lee, C.F. and Lee, J. C., "Alternative Option Pricing Models: Theory, Methods, and Applications" Kluwer Academic Publishers, to appear.
*" [http://www.rotman.utoronto.ca/%7Ehull/Technical%20Notes/TechnicalNote2.pdf Properties of Lognormal Distribution] ", John Hull, in "Options, Futures, and Other Derivatives" 6E (2005). ISBN
*Eric W. Weisstein et al. [http://mathworld.wolfram.com/LogNormalDistribution.html Log Normal Distribution] at MathWorld. Electronic document, retrieved October 26, 2006.
* Swamee, P.K. (2002). [http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JHYEFF000007000006000441000001&idtype=cvips&gifs=yes Near Lognormal Distribution] , "Journal of Hydrologic Engineering". 7(6): 441-444
*Roy B. Leipnik (1991), [http://anziamj.austms.org.au/V32/part3/Leipnik.html On Lognormal Random Variables: I - The Characteristic Function] , "Journal of the Australian Mathematical Society Series B", vol. 32, pp 327–347.

ee also

* Normal distribution
* Geometric mean
* Geometric standard deviation
* Error function
* Log-distance path loss model
* Slow fading