Log-normal distribution

Probability density function

Some log-normal density functions with identical location parameter but differing scale parameters

Cumulative distribution function

Cumulative distribution function of the log-normal distribution (with )

Parameters — location,
— scale
of associated normal
Ex. kurtosis
MGF defined only on the negative half-axis, see text
CF representation is asymptotically divergent but sufficient for numerical purposes
Fisher information

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 is log-normally distributed, then has a normal distribution. Likewise, if has a normal distribution, then 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.[1] The log-normal distribution also has been associated with other names, such as McAlister, Gibrat and Cobb–Douglas.[1]

A log-normal process is the statistical realization of 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. The log-normal distribution is the maximum entropy probability distribution for a random variate for which the mean and variance of are specified.[2]


Given a log-normally distributed random variable and two parameters and that are, respectively, the mean and standard deviation of the variable’s natural logarithm, then the logarithm of is normally distributed, and we can write as

with a standard normal variable.

This relationship is true regardless of the base of the logarithmic or exponential function. If is normally distributed, then so is , for any two positive numbers . Likewise, if is log-normally distributed, then so is , where is a positive number .

On a logarithmic scale, and 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 , s.d., and in this article. The two sets of parameters can be related as (see also Arithmetic moments below)[3]


Probability density function

A random positive variable is log-normally distributed if the logarithm of is normally distributed,

A change of variables must conserve differential probability. In particular,


is the log-normal probability density function.[1]

Cumulative distribution function

The cumulative distribution function is

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 (which can be derived by letting within the integral). However, the expected value is not defined for any positive value of the argument as the defining integral diverges. In consequence the moment generating function is not defined.[4] The last is related to the fact that the lognormal distribution is not uniquely determined by its moments.

Similarly, the characteristic function 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.[5] In particular, its Taylor formal series diverges. However, a number of alternative divergent series representations have been obtained[5][6][7][8]

A closed-form formula for the characteristic function with in the domain of convergence is not known. A relatively simple approximating formula is available in closed form and given by[9]

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


Location and scale

The location and scale parameters of a log-normal distribution, i.e. and , are more readily treated using the geometric mean, , and the geometric standard deviation, , rather than the arithmetic mean, , and the arithmetic standard deviation, .

Geometric moments

The geometric mean of the log-normal distribution is , and the geometric standard deviation is .[10][11] By analogy with the arithmetic statistics, one can define a geometric variance, , and a geometric coefficient of variation,[10] .

Because the log-transformed variable is symmetric and quantiles are preserved under monotonic transformations, the geometric mean of a log-normal distribution is equal to its median, .[12]

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,

In finance the term 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

For any real or complex number s, the s-th moment of a log-normally distributed variable X is given by[1]

Specifically, the arithmetic mean, expected square, arithmetic variance, and arithmetic standard deviation of a log-normally distributed variable X are given by


The location (μ) and scale (σ) parameters can be obtained if the arithmetic mean and the arithmetic variance are known:

A probability distribution is not uniquely determined by the moments E[Xk] = e + 1/2k2σ2 for k ≥ 1. That is, there exist other distributions with the same set of moments.[1] In fact, there is a whole family of distributions with the same moments as the log-normal distribution.

Mode and median

Comparison of mean, median and mode of two log-normal distributions with different skewness.

The mode is the point of global maximum of the probability density function. In particular, it solves the equation :

The median is such a point where :

Arithmetic coefficient of variation

The arithmetic coefficient of variation is the ratio (on the natural scale). For a log-normal distribution it is equal to

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 with respect to a threshold is defined as where is the probability density function of . Alternatively, and using the definition of conditional expectation, it can be written as . For a log-normal random variable the partial expectation is given by:

where Φ 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.

Conditional expectation

The conditional expectation of a lognormal random variable X with respect to a threshold k is its partial expectation divided by the cumulative probability of being in that range:


A set of data that arises from the log-normal distribution has a symmetric Lorenz curve (see also Lorenz asymmetry coefficient).[13]

The harmonic , geometric and arithmetic means of this distribution are related;[14] such relation is given by

Log-normal distributions are infinitely divisible,[15] but they are not stable distributions, which can be easily drawn from.[16]

Occurrence and applications

The log-normal distribution is important in the description of natural phenomena. This follows, because many natural growth processes are driven by the accumulation of many small percentage changes. These become additive on a log scale. If the effect of any one change is negligible, the central limit theorem says that the distribution of their sum is more nearly normal than that of the summands. When back-transformed onto the original scale, it makes the distribution of sizes approximately log-normal (though if the standard deviation is sufficiently small, the normal distribution can be an adequate approximation).

This multiplicative version of the central limit theorem is also known as Gibrat's law, after Robert Gibrat (1904–1980) who formulated it for companies.[17] If the rate of accumulation of these small changes does not vary over time, growth becomes independent of size. Even if that's not true, the size distributions at any age of things that grow over time tends to be log-normal.

Examples include the following:

Fitted cumulative log-normal distribution to annually maximum 1-day rainfalls, see distribution fitting
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.

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

where by we denote the probability density function of the log-normal distribution and by 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, and , reach their maximum with the same and . 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

Multivariate log-normal

If is a multivariate normal distribution then has a multivariate log-normal distribution[35][36] with mean

and covariance matrix

Related distributions

In the case that all have the same variance parameter , these formulas simplify to

This is a log-logistic distribution.

See also


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Further reading

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