WOLFRAM

Wolfram Language & System Documentation Center
Wolfram Language Home Page »

LevyDistribution
✖
LevyDistribution

✖

represents a Lévy distribution with location parameter μ and dispersion parameter σ.

Details

Background & Context

  • LevyDistribution[μ,σ] represents a statistical distribution supported on the interval and parametrized by a real number μ (called a "location parameter") and by a positive real number σ (called a "dispersion parameter"). Overall, the probability density function (PDF) of a Lévy distribution is unimodal with a single "peak" (i.e. a global maximum), though its overall shape (its height, its concentration near the line , and the horizontal location of its maximum) is determined by the values of μ and σ. In addition, the tails of the PDF are "fat" in the sense that the PDF decreases algebraically rather than exponentially for large values of . (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.) The Lévy distribution is a so-called stable distribution (StableDistribution) in the sense that any linear combination of independent Lévy-distributed random variables is again Lévy-distributed, and should not be confused with the more general terms "Pareto–Lévy distribution" and "Lévy (α)-stable distribution", which are used to describe certain subsets of stable distributions.
  • The Lévy distribution is named for French mathematician Paul Lévy, and while one can hardly separate the history of the Lévy distribution from that of the more general class of stable distributions (which were first thoroughly exposited by Lévy in the 1920s), a simplified version of the Lévy distribution PDF was first written about by Danish astronomer Holtsmark in 1919 as part of an investigation of the random fluctuations of gravitational fields. Despite initially being dismissed by mathematicians and probabilists for its apparent lack of real-world applicability, the family of stable distributions has since been employed in the analysis of a wide array of concepts including Brownian motion (cf. the so-called Lévy flight) and is now a fundamental tool in the fields of financial engineering and mathematical finance. In addition, the Lévy distribution has also been used to describe a number of important phenomena in areas such as geomagnetism, particle physics, cryptography, signal analysis, and biology.
  • RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a Lévy distribution. Distributed[x,LevyDistribution[μ,σ]], written more concisely as xLevyDistribution[μ,σ], can be used to assert that a random variable x is distributed according to a Lévy 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 Lévy distributions may be given using PDF[LevyDistribution[μ,σ],x] and CDF[LevyDistribution[μ,σ],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively, though because of its long tails, the raw and central moments (including the mean and variance) of LevyDistribution are for all and its moment-generating function (MomentGeneratingFunction) is Indeterminate.
  • DistributionFitTest can be used to test if a given dataset is consistent with a Lévy distribution, EstimatedDistribution to estimate a Lévy parametric distribution from given data, and FindDistributionParameters to fit data to a Lévy distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic Lévy distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic Lévy distribution.
  • TransformedDistribution can be used to represent a transformed Lévy 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 Lévy distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving Lévy distributions.
  • LevyDistribution is closely related to a number of other distributions. For example, LevyDistribution is an example of StableDistribution (LevyDistribution[μ,σ] is precisely StableDistribution[0,1/2,1,μ+σ,σ]) and hence is related to other stable distributions such as CauchyDistribution, LandauDistribution, and NormalDistribution. LevyDistribution is a special case of both InverseGammaDistribution and PearsonDistribution, and can be obtained as a transformation (via TransformedDistribution) of NormalDistribution. LevyDistribution is also related to HalfNormalDistribution, ChiSquareDistribution, GammaDistribution, and InverseGaussianDistribution.

Examples

open allclose all

Basic Examples  (4)Summary of the most common use cases

Probability density function:

In[4]:=4
✖
https://wolfram.com/xid/0c0rboaroxv2i-7els8w
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0c0rboaroxv2i-p20ig4
Out[5]=5
In[3]:=3
✖
https://wolfram.com/xid/0c0rboaroxv2i-x03bpv
Out[3]=3

Cumulative distribution function:

In[1]:=1
✖
https://wolfram.com/xid/0c0rboaroxv2i-0bd09t
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0c0rboaroxv2i-jw9svj
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0c0rboaroxv2i-nrj6jw
Out[3]=3

Mean and variance of a Lévy distribution are infinite:

In[1]:=1
✖
https://wolfram.com/xid/0c0rboaroxv2i-wko5m
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0c0rboaroxv2i-ikj8xg
Out[2]=2

Median:

In[1]:=1
✖
https://wolfram.com/xid/0c0rboaroxv2i-y26ogg
Out[1]=1

Scope  (6)Survey of the scope of standard use cases

Generate a sample of pseudorandom numbers from a Lévy distribution:

In[4]:=4
✖
https://wolfram.com/xid/0c0rboaroxv2i-qhtk5j

Compare its histogram to the PDF:

In[7]:=7
✖
https://wolfram.com/xid/0c0rboaroxv2i-03mwaz
Out[7]=7

Distribution parameters estimation:

In[1]:=1
✖
https://wolfram.com/xid/0c0rboaroxv2i-45b7g2

Estimate the distribution parameters from sample data:

In[2]:=2
✖
https://wolfram.com/xid/0c0rboaroxv2i-epi747
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0c0rboaroxv2i-f8ui5o
Out[3]=3

Moments of order do not exist:

In[1]:=1
✖
https://wolfram.com/xid/0c0rboaroxv2i-jgcj0z
Out[1]=1

Generalized moments exist for order :

In[2]:=2
✖
https://wolfram.com/xid/0c0rboaroxv2i-9oncw6
Out[2]=2

Hazard function:

In[1]:=1
✖
https://wolfram.com/xid/0c0rboaroxv2i-80648l
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0c0rboaroxv2i-d3g89a
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0c0rboaroxv2i-qt793f
Out[3]=3

Quantile function:

In[1]:=1
✖
https://wolfram.com/xid/0c0rboaroxv2i-wi946n
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0c0rboaroxv2i-t2pcdt
Out[2]=2

Consistent use of Quantity in parameters yields QuantityDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0c0rboaroxv2i-otote
Out[1]=1

Find quartiles:

In[2]:=2
✖
https://wolfram.com/xid/0c0rboaroxv2i-eiv1fd
Out[2]=2

Applications  (1)Sample problems that can be solved with this function

Find the full width at half-maximum of the van der Waals spectral profile:

In[1]:=1
✖
https://wolfram.com/xid/0c0rboaroxv2i-u9t9q
Out[1]=1
In[7]:=7
✖
https://wolfram.com/xid/0c0rboaroxv2i-j9unqb
Out[7]=7

Compute the location of the maximum:

In[3]:=3
✖
https://wolfram.com/xid/0c0rboaroxv2i-njhn9
Out[3]=3

Solve for the half-maximum points:

In[4]:=4
✖
https://wolfram.com/xid/0c0rboaroxv2i-io5bs
Out[4]=4

Find the width:

In[5]:=5
✖
https://wolfram.com/xid/0c0rboaroxv2i-i4fy1
Out[5]=5

The frequency of the emitted particle is more likely to be greater than the mode:

In[6]:=6
✖
https://wolfram.com/xid/0c0rboaroxv2i-zga23
Out[6]=6

Properties & Relations  (7)Properties of the function, and connections to other functions

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

In[1]:=1
✖
https://wolfram.com/xid/0c0rboaroxv2i-4mjv5h
Out[1]=1

Lévy distribution is closed under addition:

In[1]:=1
✖
https://wolfram.com/xid/0c0rboaroxv2i-th2rjz
Out[1]=1

Relationships to other distributions:

LevyDistribution[0,σ] is a special case of InverseGammaDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0c0rboaroxv2i-cs8wcv
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0c0rboaroxv2i-fn8ci7
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0c0rboaroxv2i-8d4t9a
Out[3]=3

Lévy distribution is a special case of type 5 PearsonDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0c0rboaroxv2i-3ue547
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0c0rboaroxv2i-b2jca6
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0c0rboaroxv2i-s80538
Out[3]=3

Lévy distribution is a transformation of a NormalDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0c0rboaroxv2i-k9mpt
Out[1]=1

With mean and scale:

In[2]:=2
✖
https://wolfram.com/xid/0c0rboaroxv2i-eunp24
Out[2]=2

Lévy distribution is a StableDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0c0rboaroxv2i-yulxbd
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0c0rboaroxv2i-fgdfgv
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0c0rboaroxv2i-wo7n9z
Out[3]=3

Possible Issues  (2)Common pitfalls and unexpected behavior

LevyDistribution is not defined when μ is not a real number:

In[1]:=1
✖
https://wolfram.com/xid/0c0rboaroxv2i-dh9zai
Out[1]=1

LevyDistribution is not defined when σ is not a positive real number:

In[2]:=2
✖
https://wolfram.com/xid/0c0rboaroxv2i-ebk
Out[2]=2

Substitution of invalid parameters into symbolic outputs gives results that are not meaningful:

In[1]:=1
✖
https://wolfram.com/xid/0c0rboaroxv2i-t70
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0c0rboaroxv2i-bmqlfx
Out[2]=2

Neat Examples  (1)Surprising or curious use cases

PDFs for different μ values with CDF contours:

In[1]:=1
✖
https://wolfram.com/xid/0c0rboaroxv2i-io0104
In[7]:=7
✖
https://wolfram.com/xid/0c0rboaroxv2i-ceikus
Out[7]=7
Wolfram Research (2008), LevyDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/LevyDistribution.html (updated 2016).
✖
Wolfram Research (2008), LevyDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/LevyDistribution.html (updated 2016).

Text

Wolfram Research (2008), LevyDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/LevyDistribution.html (updated 2016).

✖
Wolfram Research (2008), LevyDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/LevyDistribution.html (updated 2016).

CMS

Wolfram Language. 2008. "LevyDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/LevyDistribution.html.

✖
Wolfram Language. 2008. "LevyDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/LevyDistribution.html.

APA

Wolfram Language. (2008). LevyDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LevyDistribution.html

✖
Wolfram Language. (2008). LevyDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LevyDistribution.html

BibTeX

@misc{reference.wolfram_2025_levydistribution, author="Wolfram Research", title="{LevyDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/LevyDistribution.html}", note=[Accessed: 05-May-2025 ]}

✖
@misc{reference.wolfram_2025_levydistribution, author="Wolfram Research", title="{LevyDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/LevyDistribution.html}", note=[Accessed: 05-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_levydistribution, organization={Wolfram Research}, title={LevyDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/LevyDistribution.html}, note=[Accessed: 05-May-2025 ]}

✖
@online{reference.wolfram_2025_levydistribution, organization={Wolfram Research}, title={LevyDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/LevyDistribution.html}, note=[Accessed: 05-May-2025 ]}