Log-normal
Probability density function
Plot of the Lognormal PDF

Identical parameter but differing parameters
Cumulative distribution function
Plot of the Lognormal CDF

Notation
Parameters ,
Support
PDF
CDF
Quantile
Mean
Median
Mode
Variance
Skewness
Ex. kurtosis
Entropy
MGF defined only for numbers with a non-positive real part, see text
CF representation is asymptotically divergent but sufficient for numerical purposes
Fisher information
Method of Moments ,

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 = ln(X) has a normal distribution.[1][2] Equivalently, if Y has a normal distribution, then the exponential function of Y, X = exp(Y), has a log-normal distribution. A random variable which is log-normally distributed takes only positive real values. It is a convenient and useful model for measurements in exact and engineering sciences, as well as medicine, economics and other topics (e.g., energies, concentrations, lengths, prices of financial instruments, and other metrics).

The distribution is occasionally referred to as the Galton distribution or Galton's distribution, after Francis Galton.[3] The log-normal distribution has also been associated with other names, such as McAlister, Gibrat and Cobb–Douglas.[3]

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 (sometimes called Gibrat's law). 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 specified.[4]

Definitions

Generation and parameters

Let be a standard normal variable, and let and be two real numbers. Then, the distribution of the random variable

is called the log-normal distribution with parameters and . These are the expected value (or mean) and standard deviation of the variable's natural logarithm, not the expectation and standard deviation of itself.

Relation between normal and log-normal distribution. If 
  
    
      
        Y
        =
        μ
        +
        σ
        Z
      
    
    {\displaystyle Y=\mu +\sigma Z}
  
 is normally distributed, then 
  
    
      
        X
        ∼
        
          e
          
            Y
          
        
      
    
    {\displaystyle X\sim e^{Y))
  
 is log-normally distributed.
Relation between normal and log-normal distribution. If is normally distributed, then is log-normally distributed.

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 .

In order to produce a distribution with desired mean and variance , one uses and

Alternatively, the "multiplicative" or "geometric" parameters and can be used. They have a more direct interpretation: is the median of the distribution, and is useful for determining "scatter" intervals, see below.

Probability density function

A positive random variable X is log-normally distributed (i.e., ), if the natural logarithm of X is normally distributed with mean and variance :

Let and be respectively the cumulative probability distribution function and the probability density function of the N(0,1) distribution, then we have that[1][3]

Cumulative distribution function

The cumulative distribution function is

where is the cumulative distribution function of the standard normal distribution (i.e., N(0,1)).

This may also be expressed as follows:[1]

where erfc is the complementary error function.

Multivariate log-normal

If is a multivariate normal distribution, then has a multivariate log-normal distribution.[5][6] The exponential is applied elementwise to the random vector . The mean of is

and its covariance matrix is

Since the multivariate log-normal distribution is not widely used, the rest of this entry only deals with the univariate distribution.

Characteristic function and moment generating function

All moments of the log-normal distribution exist and

This can be derived by letting within the integral. However, the log-normal distribution is not determined by its moments.[7] This implies that it cannot have a defined moment generating function in a neighborhood of zero.[8] Indeed, the expected value is not defined for any positive value of the argument , since the defining integral diverges.

The characteristic function is defined for real values of t, but is not defined for any complex value of t that has a negative imaginary part, and hence the characteristic function is not analytic at the origin. Consequently, the characteristic function of the log-normal distribution cannot be represented as an infinite convergent series.[9] In particular, its Taylor formal series diverges:

However, a number of alternative divergent series representations have been obtained.[9][10][11][12]

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 is given by[13]

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 .

Properties

a. 
  
    
      
        y
      
    
    {\displaystyle y}
  
 is a log-normal variable with 
  
    
      
        μ
        =
        1
        ,
        σ
        =
        0.5
      
    
    {\displaystyle \mu =1,\sigma =0.5}
  
. 
  
    
      
        p
        (
        sin
        ⁡
        y
        >
        0
        )
      
    
    {\displaystyle p(\sin y>0)}
  
 is computed by transforming to the normal variable 
  
    
      
        x
        =
        ln
        ⁡
        y
      
    
    {\displaystyle x=\ln y}
  
, then integrating its density over the domain defined by 
  
    
      
        sin
        ⁡
        
          e
          
            x
          
        
        >
        0
      
    
    {\displaystyle \sin e^{x}>0}
  
 (blue regions), using the numerical method of ray-tracing.[14] b & c. The pdf and cdf of the function 
  
    
      
        sin
        ⁡
        y
      
    
    {\displaystyle \sin y}
  
 of the log-normal variable can also be computed in this way.
a. is a log-normal variable with . is computed by transforming to the normal variable , then integrating its density over the domain defined by (blue regions), using the numerical method of ray-tracing.[14] b & c. The pdf and cdf of the function of the log-normal variable can also be computed in this way.

Probability in different domains

The probability content of a log-normal distribution in any arbitrary domain can be computed to desired precision by first transforming the variable to normal, then numerically integrating using the ray-trace method.[14] (Matlab code)

Probabilities of functions of a log-normal variable

Since the probability of a log-normal can be computed in any domain, this means that the cdf (and consequently pdf and inverse cdf) of any function of a log-normal variable can also be computed.[14] (Matlab code)

Geometric or multiplicative moments

The geometric or multiplicative mean of the log-normal distribution is . It equals the median. The geometric or multiplicative standard deviation is .[15][16]

By analogy with the arithmetic statistics, one can define a geometric variance, , and a geometric coefficient of variation,[15] , has been proposed. This term was intended to be analogous to the coefficient of variation, for describing multiplicative variation in log-normal data, but this definition of GCV has no theoretical basis as an estimate of itself (see also Coefficient of variation).

Note that the geometric mean is smaller than the arithmetic mean. This is due to the AM–GM inequality and is a consequence of the logarithm being a concave function. In fact,

[17]

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 n, the n-th moment of a log-normally distributed variable X is given by[3]

Specifically, the arithmetic mean, expected square, arithmetic variance, and arithmetic standard deviation of a log-normally distributed variable X are respectively given by:[1]

The arithmetic coefficient of variation is the ratio . For a log-normal distribution it is equal to[2]

This estimate is sometimes referred to as the "geometric CV" (GCV),[18][19] due to its use of the geometric variance. Contrary to the arithmetic standard deviation, the arithmetic coefficient of variation is independent of the arithmetic mean.

The parameters μ and σ can be obtained, if the arithmetic mean and the arithmetic variance are known:

A probability distribution is not uniquely determined by the moments E[Xn] = e + 1/2n2σ2 for n ≥ 1. That is, there exist other distributions with the same set of moments.[3] In fact, there is a whole family of distributions with the same moments as the log-normal distribution.[citation needed]

Mode, median, quantiles

Comparison of mean, median and mode of two log-normal distributions with different skewness.
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, by solving the equation , we get that:

Since the log-transformed variable has a normal distribution, and quantiles are preserved under monotonic transformations, the quantiles of are

where is the quantile of the standard normal distribution.

Specifically, the median of a log-normal distribution is equal to its multiplicative mean,[20]

Partial expectation

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

Alternatively, by using the definition of conditional expectation, it can be written as . For a log-normal random variable, the partial expectation is given by:

where is the normal cumulative distribution function. The derivation of the formula is provided in the Talk page. 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 log-normal random variable —with respect to a threshold —is its partial expectation divided by the cumulative probability of being in that range:

Alternative parameterizations

In addition to the characterization by or , here are multiple ways how the log-normal distribution can be parameterized. ProbOnto, the knowledge base and ontology of probability distributions[21][22] lists seven such forms:

Overview of parameterizations of the log-normal distributions.
Overview of parameterizations of the log-normal distributions.

Examples for re-parameterization

Consider the situation when one would like to run a model using two different optimal design tools, for example PFIM[27] and PopED.[28] The former supports the LN2, the latter LN7 parameterization, respectively. Therefore, the re-parameterization is required, otherwise the two tools would produce different results.

For the transition following formulas hold and .

For the transition following formulas hold and .

All remaining re-parameterisation formulas can be found in the specification document on the project website.[29]

Multiple, reciprocal, power

Multiplication and division of independent, log-normal random variables

If two independent, log-normal variables and are multiplied [divided], the product [ratio] is again log-normal, with parameters [] and , where . This is easily generalized to the product of such variables.

More generally, if are independent, log-normally distributed variables, then

Multiplicative central limit theorem

See also: Gibrat's law

The geometric or multiplicative mean of independent, identically distributed, positive random variables shows, for approximately a log-normal distribution with parameters and , assuming is finite.

In fact, the random variables do not have to be identically distributed. It is enough for the distributions of to all have finite variance and satisfy the other conditions of any of the many variants of the central limit theorem.

This is commonly known as Gibrat's law.

Other

A set of data that arises from the log-normal distribution has a symmetric Lorenz curve (see also Lorenz asymmetry coefficient).[30]

The harmonic , geometric and arithmetic means of this distribution are related;[31] such relation is given by

Log-normal distributions are infinitely divisible,[32] but they are not stable distributions, which can be easily drawn from.[33]

Related distributions

For a more accurate approximation, one can use the Monte Carlo method to estimate the cumulative distribution function, the pdf and the right tail.[36][37]

The sum of correlated log-normally distributed random variables can also be approximated by a log-normal distribution[citation needed]

Statistical inference

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. Note that

where is the density function of the normal distribution . Therefore, the log-likelihood function is

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, the maximum likelihood estimators are identical to those for a normal distribution for the observations ,

For finite n, the estimator for is unbiased, but the one for is biased. As for the normal distribution, an unbiased estimator for can be obtained by replacing the denominator n by n−1 in the equation for .

When the individual values are not available, but the sample's mean and standard deviation s is, then the corresponding parameters are determined by the following formulas, obtained from solving the equations for the expectation and variance for and :

Statistics

The most efficient way to analyze log-normally distributed data consists of applying the well-known methods based on the normal distribution to logarithmically transformed data and then to back-transform results if appropriate.

Scatter intervals

A basic example is given by scatter intervals: For the normal distribution, the interval contains approximately two thirds (68%) of the probability (or of a large sample), and contain 95%. Therefore, for a log-normal distribution,

contains 2/3, and
contains 95% of the probability. Using estimated parameters, then approximately the same percentages of the data should be contained in these intervals.

Confidence interval for μ*

Using the principle, note that a confidence interval for is , where is the standard error and q is the 97.5% quantile of a t distribution with n-1 degrees of freedom. Back-transformation leads to a confidence interval for ,

with

Extremal principle of entropy to fix the free parameter σ

In applications, is a parameter to be determined. For growing processes balanced by production and dissipation, the use of an extremal principle of Shannon entropy shows that[41]

This value can then be used to give some scaling relation between the inflexion point and maximum point of the log-normal distribution.[41] This relationship is determined by the base of natural logarithm, , and exhibits some geometrical similarity to the minimal surface energy principle. These scaling relations are useful for predicting a number of growth processes (epidemic spreading, droplet splashing, population growth, swirling rate of the bathtub vortex, distribution of language characters, velocity profile of turbulences, etc.). For example, the log-normal function with such fits well with the size of secondarily produced droplets during droplet impact [42] and the spreading of an epidemic disease.[43]

The value is used to provide a probabilistic solution for the Drake equation.[44]

Occurrence and applications

The log-normal distribution is important in the description of natural phenomena. Many natural growth processes are driven by the accumulation of many small percentage changes which become additive on a log scale. Under appropriate regularity conditions, the distribution of the resulting accumulated changes will be increasingly well approximated by a log-normal, as noted in the section above on "Multiplicative Central Limit Theorem". This is also known as Gibrat's law, after Robert Gibrat (1904–1980) who formulated it for companies.[45] If the rate of accumulation of these small changes does not vary over time, growth becomes independent of size. Even if this assumption is not true, the size distributions at any age of things that grow over time tends to be log-normal.[citation needed] 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.[citation needed]

A second justification is based on the observation that fundamental natural laws imply multiplications and divisions of positive variables. Examples are the simple gravitation law connecting masses and distance with the resulting force, or the formula for equilibrium concentrations of chemicals in a solution that connects concentrations of educts and products. Assuming log-normal distributions of the variables involved leads to consistent models in these cases.

Specific examples are given in the following subsections.

Human behavior

Biology and medicine

Chemistry

In chemistry, the log-normal distribution is used to model particle size distributions and molar mass distributions.

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

Hydrology

The image on the right, made with CumFreq, 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.[61]
The rainfall data are represented by plotting positions as part of a cumulative frequency analysis.

Social sciences and demographics

Technology

See also

Notes

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Further reading