# Wolfram Language & System 10.4 (2016)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
BUILT-IN WOLFRAM LANGUAGE SYMBOL

# LogNormalDistribution

represents a lognormal distribution derived from a normal distribution with mean μ and standard deviation σ.

## Background & ContextBackground & Context

• represents a continuous statistical distribution supported over the interval and parametrized by a real number μ and by a positive real number σ that together determine the overall shape of its probability density function (PDF). Depending on the values of σ and μ, the PDF of a lognormal distribution may be either unimodal with a single "peak" (i.e. a global maximum) or monotone decreasing with a potential singularity approaching the lower boundary of its domain. In addition, the PDF of the lognormal distribution has tails that are "fat", in the sense that its PDF decreases algebraically rather than decreasing exponentially for large values of . (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.) The lognormal distribution is sometimes called the Galton distribution, the antilognormal distribution, or the CobbDouglas distribution.
• LogNormalDistribution is the distribution followed by the logarithm of a normally distributed random variable. In other words, if is a random variable and (where denotes "is distributed as"), then . The origins of the lognormal distribution can be traced to an observation made by Francis Galton in the 1870s demonstrating that the distribution modeling the logarithm of a product of a number of independent positive random variates tends to a standard NormalDistribution as the number of variates gets infinitely large. The theory of the distribution was studied further in the early 1900s and has since been found to accurately model both the weights of humans and the sizes of computer files on a file system. In addition, the lognormal distribution has become a widely utilized tool for modeling various phenomena, including dust concentrations, gold and uranium grades, flood flows, lifetime distributions for manufactured products, and miscellaneous phenomena in finance and economics.
• RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a lognormal distribution. Distributed[x,LogNormalDistribution[μ,σ]], written more concisely as , can be used to assert that a random variable x is distributed according to a lognormal distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
• The probability density and cumulative distribution functions for lognormal distributions may be given using PDF[LogNormalDistribution[μ,σ],x] and CDF[LogNormalDistribution[μ,σ],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively.
• DistributionFitTest can be used to test if a given dataset is consistent with a lognormal distribution, EstimatedDistribution to estimate a lognormal parametric distribution from given data, and FindDistributionParameters to fit data to a lognormal distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic lognormal distribution, and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic lognormal distribution.
• TransformedDistribution can be used to represent a transformed lognormal distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a lognormal distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving lognormal distributions.
• LogNormalDistribution is related to a number of other distributions. It can be realized as a transformation of NormalDistribution, in the sense that the PDF of TransformedDistribution[Exp[x],xNormalDistribution[μ,σ]] is precisely the same as that of . Its logarithmic behavior is qualitatively similar to that of LogLogisticDistribution, LogMultinormalDistribution, and LogGammaDistribution. LogNormalDistribution is a special case of JohnsonDistribution. One can derive SuzukiDistribution by combining LogNormalDistribution with RayleighDistribution, in the sense that the PDF of SuzukiDistribution[μ,ν] is precisely the same as that of TransformedDistribution[u v,{uRayleighDistribution[1],vLogNormalDistribution[μ,ν]}]. By way of its connection to NormalDistribution, LogNormalDistribution is also related to StableDistribution, RiceDistribution, MaxwellDistribution, LevyDistribution, LaplaceDistribution, ChiDistribution, and ChiSquareDistribution.

## ExamplesExamplesopen allclose all

### Basic Examples  (4)Basic Examples  (4)

Probability density function:

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Cumulative distribution function:

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Mean and variance:

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Median:

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