StableDistribution
✖
StableDistribution
represents the stable distribution Stype with index of stability α, skewness parameter β, location parameter μ, and scale parameter σ.
Details

- A linear combination of independent identically distributed stable random variables is also stable.
- A stable distribution is defined in terms of its characteristic function
, which satisfies a functional equation where for any
and
there exist
and
such that
. The general solution to the functional equation has four parameters.
- StableDistribution allows 0<α≤2, -1≤β≤1, μ to be any real number, and σ to be any positive real number.
- StableDistribution allows μ and σ to be any quantities of the same unit dimensions, and α, β to be dimensionless quantities. »
- CharacteristicFunction[StableDistribution[0,α,…],t] is continuous in α and given by
.
- CharacteristicFunction[StableDistribution[1,α,…],t] is discontinuous in α and given by
.
- StableDistribution[α] is equivalent to StableDistribution[1,α,0,0,1].
- StableDistribution[α,β] is equivalent to StableDistribution[1,α,β,0,1].
- StableDistribution[α,β,μ,σ] is equivalent to StableDistribution[1,α,β,μ,σ].
- StableDistribution can be used with such functions as Mean, CDF, and RandomVariate.
Background & Context
- StableDistribution[type,α,β,μ,σ] represents a continuous statistical distribution belonging to one of two types and parametrized by the positive real number σ (called a "scale parameter") and by the real numbers μ (a "location parameter"), α (the index of stability of the distribution for which
), and β (a "skewness parameter" satisfying
), which together determine the overall behavior of its probability density function (PDF).
- In general, the PDF of a stable distribution is unimodal with a single "peak" (i.e. a global maximum), though its overall shape (its height, its spread, and the horizontal location of its maximum) is determined both by its type and by the values of α, β, μ, and σ. In addition, the tails of the PDF may be "fat" (i.e. the PDF decreases non-exponentially for large values
) or "thin" (i.e. the PDF decreases exponentially for large
), depending on the values of type, α, β, μ, and σ. (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.) The four-, two-, and one-parameter versions StableDistribution[α,β,μ,σ], StableDistribution[α,β], and StableDistribution[α] are equivalent to the type-1 distributions StableDistribution[1,α,β,μ,σ], StableDistribution[1,α,β,0,1], and StableDistribution[1,α,0,0,1], respectively. For various values of its parameters, the stable distribution may be referred to as the stable-Paretian distribution, the Pareto–Lévy distribution (not to be confused with ParetoDistribution or LevyDistribution), or as the Lévy α-stable distribution. The stable distribution is also distinct from the similarly named min- and max-stable distributions (MinStableDistribution and MaxStableDistribution, respectively).
- Despite many special cases of the stable distribution being classical in nature, the family of stable distributions described above was first studied by mathematician Paul Lévy in the mid-1920s. The family of stable distributions is characterized by being closed under linear combinations, meaning their PDFs in general do not have closed-form expression and must instead be described in terms of their characteristic functions (CharacteristicFunction). Stable distributions are particularly important in stochastics and probability due to the role they play in generalizing the so-called central limit theorem, though such distributions have also been used to model phenomena in finance, astronomy, and physics.
- RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a stable distribution. Distributed[x,StableDistribution[type,α,β,μ,σ]], written more concisely as xStableDistribution[type,α,β,μ,σ], can be used to assert that a random variable x is distributed according to a stable distribution of a given type. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
- The probability density and cumulative distribution functions for stable distributions of a given type may be given using PDF[StableDistribution[type,α,β,μ,σ],x] and CDF[StableDistribution[type,α,β,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 stable distribution, EstimatedDistribution to estimate a stable parametric distribution from given data, and FindDistributionParameters to fit data to a stable distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic stable distribution, and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic stable distribution.
- TransformedDistribution can be used to represent a transformed stable 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 stable distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving stable distributions.
- StableDistribution is closely related to a number of other distributions. LandauDistribution, CauchyDistribution, NormalDistribution, and LevyDistribution are examples of type-1 stable distributions in the sense that LandauDistribution[μ,σ] has the same characteristic function (CharacteristicFunction) as StableDistribution[1,1,1,μ,σ], CauchyDistribution[0,1] has the same PDF as StableDistribution[1,1,0,0,1], the PDF of NormalDistribution[μ,σ] is equivalent to that of StableDistribution[1,2,β,μ,σ/
], and the PDF of LevyDistribution[μ,σ] is precisely the same as that of StableDistribution[1,1/2,1,μ,σ]. Qualitatively, StableDistribution is similar to PearsonDistribution, and it is also closely related to ParetoDistribution, BetaDistribution, GammaDistribution, and HalfNormalDistribution.
Examples
open allclose allBasic Examples (4)Summary of the most common use cases
Probability density function for type 1 for a range of skewness parameters:

https://wolfram.com/xid/0bnhgp071je6i-61u8pd

Probability density function for type 0 for various stability indexes:

https://wolfram.com/xid/0bnhgp071je6i-22y84s

Cumulative distribution function for type 1:

https://wolfram.com/xid/0bnhgp071je6i-0k63lk


https://wolfram.com/xid/0bnhgp071je6i-yc7jct


https://wolfram.com/xid/0bnhgp071je6i-1rsxul


https://wolfram.com/xid/0bnhgp071je6i-9imddx

Variance is type independent and is only defined for :

https://wolfram.com/xid/0bnhgp071je6i-2hgob1


https://wolfram.com/xid/0bnhgp071je6i-y4ayui

Scope (4)Survey of the scope of standard use cases
Generate a sample of pseudorandom numbers from a stable distribution:

https://wolfram.com/xid/0bnhgp071je6i-tscvhm
Compare its histogram to the PDF:

https://wolfram.com/xid/0bnhgp071je6i-fw0ala

Higher moments are only defined for :

https://wolfram.com/xid/0bnhgp071je6i-lpkqey

This is the case where StableDistribution reduces to NormalDistribution:

https://wolfram.com/xid/0bnhgp071je6i-c1ic0u

Hazard function for different stability indexes:

https://wolfram.com/xid/0bnhgp071je6i-rwxqe

Hazard function for different skewness parameters:

https://wolfram.com/xid/0bnhgp071je6i-i4zm47

Consistent use of Quantity in parameters yields QuantityDistribution:

https://wolfram.com/xid/0bnhgp071je6i-bsnti9


https://wolfram.com/xid/0bnhgp071je6i-gpku4f

Applications (9)Sample problems that can be solved with this function
Assuming daily logarithmic return of the stock market follows a stable distribution, simulate and visualize stock prices over a period of 5 years:

https://wolfram.com/xid/0bnhgp071je6i-lkl8l5

https://wolfram.com/xid/0bnhgp071je6i-wb23d

Assuming stock logarithmic return follows a stable distribution, find the value at risk at the 95% level:

https://wolfram.com/xid/0bnhgp071je6i-k7g4dd

https://wolfram.com/xid/0bnhgp071je6i-czspj

Compute the 95% value at risk point loss of the current S&P 500 index value, assuming the above distribution:

https://wolfram.com/xid/0bnhgp071je6i-4lmkm

Find the expected shortfall of logarithmic return:

https://wolfram.com/xid/0bnhgp071je6i-c0tv9a

Compute the associated point loss:

https://wolfram.com/xid/0bnhgp071je6i-c6jv19

Fit the daily logarithm return of the IBM stocks since January 1, 2005 to a stable distribution:

https://wolfram.com/xid/0bnhgp071je6i-s09vhx

https://wolfram.com/xid/0bnhgp071je6i-b4vvm

https://wolfram.com/xid/0bnhgp071je6i-edue1i


https://wolfram.com/xid/0bnhgp071je6i-ktsk4f
Fit logarithmic returns to a stable distribution:

https://wolfram.com/xid/0bnhgp071je6i-jzpyl8

Compare the estimated distribution to a data histogram:

https://wolfram.com/xid/0bnhgp071je6i-fhjcvg

The product of a symmetric stable random variate and the power of an exponential random variate follows a Linnik distribution:

https://wolfram.com/xid/0bnhgp071je6i-ijc04v
Calculate the characteristic function of a Linnik distribution:

https://wolfram.com/xid/0bnhgp071je6i-cwesfy


https://wolfram.com/xid/0bnhgp071je6i-gdbs2u

Generate random variates and show the histogram:

https://wolfram.com/xid/0bnhgp071je6i-fqw4d7

Map–Airy distribution [MathWorld] is a member of the stable family:

https://wolfram.com/xid/0bnhgp071je6i-gmjmf1
Its probability density is known in closed form:

https://wolfram.com/xid/0bnhgp071je6i-d9dqr0


https://wolfram.com/xid/0bnhgp071je6i-jq38zm

Find the location of the mode:

https://wolfram.com/xid/0bnhgp071je6i-db1d7g

Estimate the parameter of stable distribution from a sample characteristic function:

https://wolfram.com/xid/0bnhgp071je6i-mz1ysf

https://wolfram.com/xid/0bnhgp071je6i-pug6py

https://wolfram.com/xid/0bnhgp071je6i-d1bcov

Plot absolute values of a sample characteristic function and a population characteristic function:

https://wolfram.com/xid/0bnhgp071je6i-0l8i7


https://wolfram.com/xid/0bnhgp071je6i-by8u9u

Compare with the maximum likelihood estimation:

https://wolfram.com/xid/0bnhgp071je6i-n2cz0t

Generalized central limit theorem gives sequences and
such that the distribution of the shifted and rescaled sum
of
i.i.d. random variates
whose distribution function
has asymptotes
as
and
as
weakly converges to the stable distribution
:

Illustrate the generalized central limit theorem using a two-sided Pareto distribution:

https://wolfram.com/xid/0bnhgp071je6i-sq9b4
Define the mean and variance of the two-sided Pareto distribution for future use:

https://wolfram.com/xid/0bnhgp071je6i-bsm74x

https://wolfram.com/xid/0bnhgp071je6i-bn85jc
Define a routine to generate
-variates:

https://wolfram.com/xid/0bnhgp071je6i-gezeu4
Define a function to visualize a density plot and data histogram:

https://wolfram.com/xid/0bnhgp071je6i-elofwm

https://wolfram.com/xid/0bnhgp071je6i-phvlws

https://wolfram.com/xid/0bnhgp071je6i-qdzygf


https://wolfram.com/xid/0bnhgp071je6i-d14fbw

https://wolfram.com/xid/0bnhgp071je6i-nnv9t8


https://wolfram.com/xid/0bnhgp071je6i-exvt0g

https://wolfram.com/xid/0bnhgp071je6i-jpzlvt


https://wolfram.com/xid/0bnhgp071je6i-b6w3ma

https://wolfram.com/xid/0bnhgp071je6i-xacmu

Case of standard central limit theorem, i.e. :

https://wolfram.com/xid/0bnhgp071je6i-hph37q

https://wolfram.com/xid/0bnhgp071je6i-e1cdcu

Holtsmark distribution is the distribution of forces acting on a particle in an infinite Poisson system. The ‐component of the gravitational force follows symmetric stable distribution:

https://wolfram.com/xid/0bnhgp071je6i-xs481
Simulate absolute value of the force:

https://wolfram.com/xid/0bnhgp071je6i-d4rcpp

Distribution of the absolute value is known in closed form:

https://wolfram.com/xid/0bnhgp071je6i-g326sw


https://wolfram.com/xid/0bnhgp071je6i-f750vn

A maximally right‐skewed stable variate, i.e. , is often referred to as positive stable:

https://wolfram.com/xid/0bnhgp071je6i-h2j68t
The Laplace transform of a positive stable density has the simple form :

https://wolfram.com/xid/0bnhgp071je6i-izny6k


https://wolfram.com/xid/0bnhgp071je6i-m5uzpa

Properties & Relations (11)Properties of the function, and connections to other functions
Stable variables of type 0 and type 1 are related to each other by a shift of location parameter:

https://wolfram.com/xid/0bnhgp071je6i-ciyl55
Verify using characteristic function:

https://wolfram.com/xid/0bnhgp071je6i-f5rro3


https://wolfram.com/xid/0bnhgp071je6i-bcaisu

Discontinuity in of an
stable random variate is manifested in the sensitivity of mode to small changes in
:

https://wolfram.com/xid/0bnhgp071je6i-bzsp8t

A family of stable distributions of type 0 is closed under shifting and scaling:

https://wolfram.com/xid/0bnhgp071je6i-zggnk

The proof uses characteristic functions:

https://wolfram.com/xid/0bnhgp071je6i-2zpwj8


https://wolfram.com/xid/0bnhgp071je6i-l3d6ze


https://wolfram.com/xid/0bnhgp071je6i-bcsmhy

A family of stable distributions of type 1 is closed under shifting and scaling:

https://wolfram.com/xid/0bnhgp071je6i-nugwvo


https://wolfram.com/xid/0bnhgp071je6i-zxuzlv

Sum of two stable variates of the same stability index is again a stable variate:

https://wolfram.com/xid/0bnhgp071je6i-j6wn66

https://wolfram.com/xid/0bnhgp071je6i-koa49i
Considering first the case when is not 1:

https://wolfram.com/xid/0bnhgp071je6i-cygyk3


https://wolfram.com/xid/0bnhgp071je6i-fe2r2x

Strictly stable distributions satisfy duality law, for and
:

https://wolfram.com/xid/0bnhgp071je6i-btsal4
Dual strictly stable distribution with stability index :

https://wolfram.com/xid/0bnhgp071je6i-bjf76w
The duality law states that for , the following equality holds:

https://wolfram.com/xid/0bnhgp071je6i-lous53

Stable variates with are stochastically ordered, i.e.
implies
:

https://wolfram.com/xid/0bnhgp071je6i-dxar4h

Relationships to other distributions:

LandauDistribution is a stable distribution:

https://wolfram.com/xid/0bnhgp071je6i-6slhh0


https://wolfram.com/xid/0bnhgp071je6i-i86myc


https://wolfram.com/xid/0bnhgp071je6i-7q2hyd

CauchyDistribution is a stable distribution:

https://wolfram.com/xid/0bnhgp071je6i-zkvczc


https://wolfram.com/xid/0bnhgp071je6i-d6bsov


https://wolfram.com/xid/0bnhgp071je6i-bum801

NormalDistribution is a stable distribution:

https://wolfram.com/xid/0bnhgp071je6i-xkrde7


https://wolfram.com/xid/0bnhgp071je6i-49eymo


https://wolfram.com/xid/0bnhgp071je6i-nuxytu

LevyDistribution is a stable distribution:

https://wolfram.com/xid/0bnhgp071je6i-yulxbd


https://wolfram.com/xid/0bnhgp071je6i-fgdfgv


https://wolfram.com/xid/0bnhgp071je6i-wo7n9z

Wolfram Research (2010), StableDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/StableDistribution.html (updated 2016).
Text
Wolfram Research (2010), StableDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/StableDistribution.html (updated 2016).
Wolfram Research (2010), StableDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/StableDistribution.html (updated 2016).
CMS
Wolfram Language. 2010. "StableDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/StableDistribution.html.
Wolfram Language. 2010. "StableDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/StableDistribution.html.
APA
Wolfram Language. (2010). StableDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/StableDistribution.html
Wolfram Language. (2010). StableDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/StableDistribution.html
BibTeX
@misc{reference.wolfram_2025_stabledistribution, author="Wolfram Research", title="{StableDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/StableDistribution.html}", note=[Accessed: 19-May-2025
]}
BibLaTeX
@online{reference.wolfram_2025_stabledistribution, organization={Wolfram Research}, title={StableDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/StableDistribution.html}, note=[Accessed: 19-May-2025
]}