# Log-normal distribution

In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable $X$ is log-normally distributed, then $Y=\log(X)$ has a normal distribution. Likewise, if $Y$ has a normal distribution, then $X=\exp(Y)$ has a log-normal distribution. A random variable which is log-normally distributed takes only positive real values.

The distribution is occasionally referred to as the Galton distribution or Galton's distribution, after Francis Galton. The log-normal distribution also has been associated with other names, such as McAlister, Gibrat and Cobb–Douglas.

A variable might be modeled as log-normal if it can be thought of as the multiplicative product of many independent random variables 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 discount factor can be derived from the product of short-term discount factors. In wireless communication, the delay caused by shadowing or slow fading from random objects is often assumed to be log-normally distributed: see log-distance path loss model.

The log-normal distribution is the maximum entropy probability distribution for a random variate X for which the mean and variance of $\ln(X)$ are fixed.

## Notation

Given a log-normally distributed random variable X and two parameters μ and σ that are, respectively, the mean and standard deviation of the variable’s natural logarithm (by definition, the variable’s logarithm is normally distributed), we can write X as

$X=e^{\mu +\sigma Z}$ with Z a standard normal variable.

This relationship is true regardless of the base of the logarithmic or exponential function. If loga(Y) is normally distributed, then so is logb(Y), for any two positive numbers ab ≠ 1. Likewise, if $e^{X}$ is log-normally distributed, then so is $a^{X}$ , where $a$ is a positive number ≠ 1.

On a logarithmic scale, μ and σ can be called the location parameter and the scale parameter, respectively.

In contrast, the mean, standard deviation, and variance of the non-logarithmized sample values are respectively denoted m, s.d., and v in this article. The two sets of parameters can be related as (see also Arithmetic moments below)

$\mu =\ln \left({\frac {m^{2}}{\sqrt {v+m^{2}}}}\right),\sigma ={\sqrt {\ln \left(1+{\frac {v}{m^{2}}}\right)}}$ .

## Characterization

### Probability density function

The probability density function of a log-normal distribution is:

$f_{X}(x;\mu ,\sigma )={\frac {1}{x\sigma {\sqrt {2\pi }}}}\,e^{-{\frac {(\ln x-\mu )^{2}}{2\sigma ^{2}}}},\ \ x>0$ This follows by applying the change-of-variables rule on the density function of a normal distribution.

### Cumulative distribution function

$F_{X}(x;\mu ,\sigma )={\frac {1}{2}}\left[1+\operatorname {erf} \!\left({\frac {\ln x-\mu }{\sigma {\sqrt {2}}}}\right)\right]={\frac {1}{2}}\operatorname {erfc} \!\left(-{\frac {\ln x-\mu }{\sigma {\sqrt {2}}}}\right)=\Phi {\bigg (}{\frac {\ln x-\mu }{\sigma }}{\bigg )},$ where erfc is the complementary error function, and Φ is the cumulative distribution function of the standard normal distribution.

### Characteristic function and moment generating function

All moments of the log-normal distribution exist and it holds that: $\operatorname {E} (X^{n})=\mathrm {e} ^{n\mu +{\frac {n^{2}\sigma ^{2}}{2}}}$ (which can be derived by letting $z={\frac {\ln(x)-(\mu +n\sigma ^{2})}{\sigma }}$ within the integral). However, the expected value $\operatorname {E} (e^{tX})$ is not defined for any positive value of the argument $t$ as the defining integral diverges. In consequence the moment generating function is not defined. The last is related to the fact that the lognormal distribution is not uniquely determined by its moments.

Similarly, the characteristic function E[e itX] is not defined in the half complex plane and therefore it is not analytic in the origin. In consequence, the characteristic function of the log-normal distribution cannot be represented as an infinite convergent series. In particular, its Taylor formal series $\sum _{n=0}^{\infty }{\frac {(it)^{n}}{n!}}e^{n\mu +n^{2}\sigma ^{2}/2}$ diverges. However, a number of alternative divergent series representations have been obtained

A closed-form formula for the characteristic function $\varphi (t)$ with $t$ in the domain of convergence is not known. A relatively simple approximating formula is available in closed form and given by

where $W$ is the Lambert W function. This approximation is derived via an asymptotic method but it stays sharp all over the domain of convergence of $\varphi$ .

## Properties

### Location and scale

#### Geometric moments

Because the log-transformed variable $Y=\ln X$ is symmetric and quantiles are preserved under monotonic transformations, the geometric mean of a log-normal distribution is equal to its median, $\mathrm {Med} [X]$ .

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. In fact,

{\begin{aligned}\mathrm {E} [X]&=e^{\mu +{\frac {1}{2}}\sigma ^{2}}&=e^{\mu }\cdot {\sqrt {e^{\sigma ^{2}}}}&=\mathrm {GM} [X]\cdot {\sqrt {\mathrm {GVar} [X]}}.\end{aligned}} In finance the term $e^{-{\frac {1}{2}}\sigma ^{2}}$ is sometimes interpreted as a convexity correction. From the point of view of stochastic calculus, this is the same correction term as in Itō's lemma for geometric Brownian motion.

#### Arithmetic moments

The arithmetic mean, arithmetic variance, and arithmetic standard deviation of a log-normally distributed variable $X$ are given by

{\begin{aligned}&\operatorname {E} [X]=e^{\mu +{\tfrac {1}{2}}\sigma ^{2}},\\&\operatorname {Var} [X]=(e^{\sigma ^{2}}-1)e^{2\mu +\sigma ^{2}}=(e^{\sigma ^{2}}-1)(\operatorname {E} [X])^{2},\\&\operatorname {SD} [X]={\sqrt {\operatorname {Var} [X]}}=e^{\mu +{\tfrac {1}{2}}\sigma ^{2}}{\sqrt {e^{\sigma ^{2}}-1}}=\operatorname {E} [X]{\sqrt {e^{\sigma ^{2}}-1}},\end{aligned}} respectively.

The location ($\mu$ ) and scale ($\sigma$ ) parameters can be obtained if the arithmetic mean and the arithmetic variance are known; it is simpler if $\sigma$ is computed first:

{\begin{aligned}\mu &=\ln(\operatorname {E} [X])-{\frac {1}{2}}\ln \!\left(1+{\frac {\mathrm {Var} [X]}{(\operatorname {E} [X])^{2}}}\right)=\ln(\operatorname {E} [X])-{\frac {1}{2}}\sigma ^{2},\\\sigma ^{2}&=\ln \!\left(1+{\frac {\operatorname {Var} [X]}{(\operatorname {E} [X])^{2}}}\right).\end{aligned}} $\operatorname {E} [X^{s}]=e^{s\mu +{\frac {1}{2}}s^{2}\sigma ^{2}}.$ A log-normal distribution is not uniquely determined by its moments E[Xk] for k ≥ 1, that is, there exists some other distribution with the same moments for all k. In fact, there is a whole family of distributions with the same moments as the log-normal distribution.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }} ### Mode and median The mode is the point of global maximum of the probability density function. In particular, it solves the equation (ln ƒ)′ = 0: $\mathrm {Mode} [X]=e^{\mu -\sigma ^{2}}.$ The median is such a point where FX = 1/2: $\mathrm {Med} [X]=e^{\mu }\,.$ ### Arithmetic coefficient of variation $\mathrm {CV} [X]={\sqrt {e^{\sigma ^{2}}-1}}.$ Contrary to the arithmetic standard deviation, the arithmetic coefficient of variation is independent of the arithmetic mean. ### 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 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: $g(k)=\int _{k}^{\infty }\!xf(x)\,dx=e^{\mu +{\tfrac {1}{2}}\sigma ^{2}}\,\Phi \!\left({\frac {\mu +\sigma ^{2}-\ln k}{\sigma }}\right).$ Where Phi is the normal cumulative distribution function. 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. ### Other A set of data that arises from the log-normal distribution has a symmetric Lorenz curve (see also Lorenz asymmetry coefficient). The harmonic (H), geometric (G) and arithmetic (A) means of this distribution are related; such relation is given by $H={\frac {G^{2}}{A}}.$ Log-normal distributions are infinitely divisible. ## Occurrence The log-normal distribution is important in the description of natural phenomena. 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 companies. It can be shown that a growth process following Gibrat's law will result in entity sizes with a log-normal distribution. Examples include: • In biology and medicine, • Measures of size of living tissue (length, skin area, weight); • 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. • The length of inert appendages (hair, claws, nails, teeth) of biological specimens, in the direction of growth;{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B=

{{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}

• Certain physiological measurements, such as blood pressure of adult humans (after separation on male/female subpopulations)
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.
• 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.

## 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

$f_{L}(x;\mu ,\sigma )=\prod _{i=1}^{n}\left({\frac {1}{x}}_{i}\right)\,f_{N}(\ln x;\mu ,\sigma )$ where by ƒL we denote the probability density function of the log-normal distribution and by ƒN that of the normal distribution. Therefore, using the same indices to denote distributions, we can write the log-likelihood function thus:

{\begin{aligned}\ell _{L}(\mu ,\sigma |x_{1},x_{2},\dots ,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{aligned}} Since the first term is constant with regard to μ and σ, both logarithmic likelihood functions, L and 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

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

$\operatorname {E} [{\boldsymbol {Y}}]_{i}=e^{\mu _{i}+{\frac {1}{2}}\Sigma _{ii}},$ $\operatorname {Var} [{\boldsymbol {Y}}]_{ij}=e^{\mu _{i}+\mu _{j}+{\frac {1}{2}}(\Sigma _{ii}+\Sigma _{jj})}(e^{\Sigma _{ij}}-1).$ ## Generating log-normally distributed random variates

Given a random variate Z drawn from the normal distribution with 0 mean and 1 standard deviation, then the variate

$X=e^{\mu +\sigma Z}\,$ ## Related distributions

$Y\sim \operatorname {Log-{\mathcal {N}}} {\Big (}\textstyle \sum _{j=1}^{n}\mu _{j},\ \sum _{j=1}^{n}\sigma _{j}^{2}{\Big )}.$ {\begin{aligned}\sigma _{Z}^{2}&=\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],\\\mu _{Z}&=\log \!\left[\sum e^{\mu _{j}+\sigma _{j}^{2}/2}\right]-{\frac {\sigma _{Z}^{2}}{2}}.\end{aligned}} {\begin{aligned}\sigma _{Z}^{2}&=\log \!\left[(e^{\sigma ^{2}}-1){\frac {\sum e^{2\mu _{j}}}{(\sum e^{\mu _{j}})^{2}}}+1\right],\\\mu _{Z}&=\log \!\left[\sum e^{\mu _{j}}\right]+{\frac {\sigma ^{2}}{2}}-{\frac {\sigma _{Z}^{2}}{2}}.\end{aligned}} ## Similar distributions

A substitute for the log-normal whose integral can be expressed in terms of more elementary functions can be obtained based on the logistic distribution to get an approximation for the CDF

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