represents a Moyal distribution with location parameter μ and scale parameter σ.


represents a Moyal distribution with location parameter 0 and scale parameter 1.


Background & Context

  • MoyalDistribution[μ,σ] represents a continuous statistical distribution defined and supported over the set of real numbers and parametrized by a real number μ (called a "location parameter") and by a positive real number σ (called a "scale parameter"). Together, these parameters determine the overall behavior of the probability density function (PDF). In general, the PDF of a Moyal distribution is unimodal with a single "peak" (i.e. a global maximum), though its overall shape (its height, spread, and horizontal location of its maximum) is determined by the values of μ and σ. In addition, the tails of the PDF are "thin" in the sense that the PDF decreases exponentially rather than decreasing algebraically for large values of . (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.) The zero-argument form MoyalDistribution[] is equivalent to MoyalDistribution[0,1] and is sometimes referred to as the standard Moyal distribution.
  • The Moyal distribution was first proposed in a 1955 paper by physicist J. E. Moyal describing statistical methods in quantum mechanics. The distribution models the energy lost by a fast charged particle (and hence the number of ion pairs produced) during ionization. Historically, the Moyal distribution has been utilized in the approximation of the LandauDistribution and has since found use in modeling a wide array of phenomena including airline spills and the dynamics of chain-like bodies.
  • RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a Moyal distribution. Distributed[x,MoyalDistribution[μ,σ]], written more concisely as xMoyalDistribution[μ,σ], can be used to assert that a random variable x is distributed according to a Moyal 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 Moyal distributions may be given using PDF[MoyalDistribution[μ,σ],x] and CDF[MoyalDistribution[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 Moyal distribution, EstimatedDistribution to estimate a Moyal parametric distribution from given data, and FindDistributionParameters to fit data to a Moyal distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic Moyal distribution, and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic Moyal distribution.
  • TransformedDistribution can be used to represent a transformed Moyal 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 Moyal distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving Moyal distributions.
  • MoyalDistribution is related to a number of other distributions. As mentioned above, the qualitative similarities between MoyalDistribution and LandauDistribution make the PDF of the former a worthwhile tool for approximating the latter. Moreover, MoyalDistribution can be obtained as a transformation of both GammaDistribution and ExpGammaDistribution in the sense that the PDF of MoyalDistribution[μ,σ] is the distribution of both μ-σ*Log[u] when uGammaDistribution[1/2,2] (for σ>0) and -u when uExpGammaDistribution[1/2,σ,-μ+σ*Log[2]]. MoyalDistribution is also related to CauchyDistribution, LevyDistribution, and StableDistribution.


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Basic Examples  (4)

Probability density function:

Cumulative distribution function:

Mean and variance:


Scope  (7)

Generate a sample of pseudorandom numbers from a Moyal distribution:

Compare its histogram to the PDF:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare a density histogram of the sample with the PDF of the estimated distribution:

Skewness and kurtosis are constant:

Different moments with closed forms as functions of parameters:





Closed form for symbolic order:

Hazard function:

Quantile function:

Consistent use of Quantity in parameters yields QuantityDistribution:

Find the 75% quantile:

Applications  (2)

MoyalDistribution was obtained as a steepest descent approximation to LandauDistribution:

Find half-width of MoyalDistribution:

Properties & Relations  (4)

Moyal distribution is closed under translation and scaling by a positive factor:

Relationships to other distributions:

Moyal distribution is a transformation of a GammaDistribution:

Moyal distribution is a transformed ExpGammaDistribution:

Neat Examples  (1)

PDFs for different σ values with CDF contours:

Wolfram Research (2010), MoyalDistribution, Wolfram Language function, (updated 2016).


Wolfram Research (2010), MoyalDistribution, Wolfram Language function, (updated 2016).


Wolfram Language. 2010. "MoyalDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016.


Wolfram Language. (2010). MoyalDistribution. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_moyaldistribution, author="Wolfram Research", title="{MoyalDistribution}", year="2016", howpublished="\url{}", note=[Accessed: 23-June-2024 ]}


@online{reference.wolfram_2024_moyaldistribution, organization={Wolfram Research}, title={MoyalDistribution}, year={2016}, url={}, note=[Accessed: 23-June-2024 ]}