Log-normal distribution

Log-normal

Probability density function μ=0
Cumulative distribution function μ=0
Parameters	$\sigma \geq 0$ $-\infty \leq \mu \leq \infty$
Support	$x\in [0;+\infty )\!$
PDF	$\exp \left(-\left.\left[{\frac {\ln(x)-\mu }{\sigma }}\right]^{2}\right/2\right)\left/\left(x\sigma {\sqrt {2\pi }}\right)\right.$
CDF	${\frac {1}{2}}+{\frac {1}{2}}\mathrm {erf} \left[{\frac {\ln(x)-\mu }{\sigma {\sqrt {2}}}}\right]$
Mean	$e^{\mu +\sigma ^{2}/2}$
Median	$e^{\mu }$
Mode	$e^{\mu -\sigma ^{2}}$
Variance	$(e^{\sigma ^{2}}\!\!-1)e^{2\mu +\sigma ^{2}}$
Skewness	$(e^{\sigma ^{2}}\!\!+2){\sqrt {e^{\sigma ^{2}}\!\!-1}}$
Excess kurtosis	$e^{4\sigma ^{2}}\!\!+2e^{3\sigma ^{2}}\!\!+3e^{2\sigma ^{2}}\!\!-6$
Entropy	${\frac {1}{2}}+{\frac {1}{2}}\ln(2\pi \sigma ^{2})+\mu$
MGF	(see text for raw moments)

In probability and statistics, the log-normal distribution is the probability distribution of any random variable whose logarithm is normally distributed. If X is a random variable with a normal distribution, then exp(X) has a log-normal distribution; likewise, if Y is log-normally distributed, then log(Y) is normally distributed.

"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 small independent factors. A typical example is the long-term return rate on a stock investment: it can be considered as the product of the daily return rates.

The log-normal distribution has the probability density function (pdf)

$f(x;\mu ,\sigma )={\frac {e^{-(\ln x-\mu )^{2}/(2\sigma ^{2})}}{x\sigma {\sqrt {2\pi }}}}$

for $x>0$, where $\mu$ and $\sigma$ are the mean and standard deviation of the variable's logarithm. The expected value is

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

and the variance is

$\mathrm {var} (X)=(e^{\sigma ^{2}}-1)e^{2\mu +\sigma ^{2}}.\,$

Equivalent relationships may be written to obtain $\mu$ and $\sigma$ given the expected value and standard deviation:

$\mu =\ln(\mathrm {E} (X))-{\frac {1}{2}}\ln \left(1+{\frac {\mathrm {var} (X)}{(\mathrm {E} X)^{2}}}\right),$

$\sigma ^{2}=\ln \left(1+{\frac {\mathrm {var} (X)}{(\mathrm {E} X)^{2}}}\right).$

Geometric mean and geometric standard deviation

The geometric mean of the log-normal distribution is exp(μ) and, and the geometric standard deviation is equal to exp(σ).

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.

Confidence interval bounds	log space	geometric
3σ lower bound	$\mu -3\sigma$	$\mu _{\mathrm {geo} }/\sigma _{\mathrm {geo} }^{3}$
2σ lower bound	$\mu -2\sigma$	$\mu _{\mathrm {geo} }/\sigma _{\mathrm {geo} }^{2}$
1σ lower bound	$\mu -\sigma$	$\mu _{\mathrm {geo} }/\sigma _{\mathrm {geo} }$
1σ upper bound	$\mu +\sigma$	$\mu _{\mathrm {geo} }\sigma _{\mathrm {geo} }$
2σ upper bound	$\mu +2\sigma$	$\mu _{\mathrm {geo} }\sigma _{\mathrm {geo} }^{2}$
3σ upper bound	$\mu +3\sigma$	$\mu _{\mathrm {geo} }\sigma _{\mathrm {geo} }^{3}$

Where geometric mean $\mu _{\mathrm {geo} }=\exp(\mu )$ and geometric standard deviation $\sigma _{\mathrm {geo} }=\exp(\sigma )$

Moments

The first few raw moments are:

$\mu _{1}=e^{\mu +\sigma ^{2}/2}$

$\mu _{2}=e^{2\mu +4\sigma ^{2}/2}$

$\mu _{3}=e^{3\mu +9\sigma ^{2}/2}$

$\mu _{4}=e^{4\mu +16\sigma ^{2}/2}$

or generally:

$\mu _{k}=e^{k\mu +k^{2}\sigma ^{2}/2}.$

Partial expectation

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

$g(k)=\int _{k}^{\infty }xf(x)\,dx$

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

$g(k)=\exp(\mu +\sigma ^{2}/2)\Phi \left({\frac {-\ln(k)+\mu +\sigma ^{2}}{\sigma }}\right)$

where $\Phi$ 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 $f_{L}(\cdot )$ we denote the density probability function of the log-normal distribution and by $f_{N}(\cdot )$—that of the normal distribution. Therefore, using the same indices to denote distributions, we can write the log-likelihood function thus:

${\begin{matrix}\ell _{L}(\mu ,\sigma |x_{1},x_{2},...,x_{n})&=&-\sum _{k}\ln x_{k}+\ell _{N}(\mu ,\sigma |\ln x_{1},\ln x_{2},\dots ,\ln x_{n})=\\\\\ &=&\operatorname {constant} +\ell _{N}(\mu ,\sigma |\ln x_{1},\ln x_{2},\dots ,\ln x_{n}).\end{matrix}}$

Since the first term is constant with regards 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

${\widehat {\mu }}={\frac {\sum _{k}\ln x_{k}}{n}},\ {\widehat {\sigma }}^{2}={\frac {\sum _{k}{\left(\ln x_{k}-{\widehat {\mu }}\right)^{2}}}{n}}.$

Related distributions

• $Y\sim N(\mu ,\sigma ^{2})$ is a normal distribution if $Y=\ln(X)$ and $X\sim \operatorname {Log-N} (\mu ,\sigma ^{2})$.
• If $X_{m}\sim \operatorname {Log-N} (\mu ,\sigma _{m}^{2}),\ m={\overline {1...n}}$ 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: $Y\sim \operatorname {Log-N} \left(n\mu ,\sum _{m=1}^{n}\sigma _{m}^{2}\right)$.
• Let $X_{m}\sim \operatorname {Log-N} (\mu _{m},\sigma ^{2}),\ m={1,...,n}\$ be independent log-normally distributed variables with the

same $\sigma$ parameter and possibly varying $\mu$ parameters, and $Y=\sum _{m=1}^{n}X_{m}$. The distribution of $Y$ has no closed-form expression, but can be reasonably approximated by another log-normal distribution $Z$. A simple approximation is obtained by matching the mean and variance:

$\sigma _{Z}^{2}=\log \left[(e^{\sigma ^{2}}-1){\frac {\sum e^{2\mu _{m}}}{(\sum e^{\mu _{m}})^{2}}}+1\right]$

$\mu _{Z}={\frac {\sigma _{Z}^{2}}{2}}-{\frac {\sigma ^{2}}{2}}+\log \left(\sum e^{2\mu _{m}}\right)/2$

(Note: Use with caution. Formulas were copied from lecture notes found on the web which did not include the "/2" in the formula for $\mu _{Z}$; however without this the formula is inconsistent for $m=1$.)

• 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

$F(x;\mu ,\sigma )=\left[\left({\frac {e^{\mu }}{x}}\right)^{\pi /(\sigma {\sqrt {3}})}+1\right]^{-1}.$

Further reading

• Robert Brooks, Jon Corson, and J. Donal Wales. "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)
• Log-normal Distributions across the Sciences: Keys and Clues, E. Limpert, W. Stahel and M. Abbt,. BioScience, 51 (5), p. 341–352 (2001).
• 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.
• Properties of Lognormal Distribution, John Hull, in Options, Futures, and Other Derivatives 6E (2005). ISBN 0-13-149908-4
• Eric W. Weisstein et al. Log Normal Distribution at MathWorld. Electronic document, retrieved October 26, 2006.
• Swamee, P.K. (2002). Near Lognormal Distribution, Journal of Hydrologic Engineering. 6(1): 441-444

See also

• Geometric mean
• Geometric standard deviation
• Error function