PearsonDistribution
✖
PearsonDistribution
represents a distribution of the Pearson family with parameters a1, a0, b2, b1, and b0.
Details

- The probability density
satisfies the differential equation
.
- The Pearson family of distributions is historically divided into seven types. By giving the form PearsonDistribution[type,…], the type will implicitly provide domain and parameter constraints.
- PearsonDistribution[1,…] is a shifted and rescaled BetaDistribution.
- PearsonDistribution[2,…] is a symmetric shifted and rescaled BetaDistribution.
- PearsonDistribution[3,…] includes NormalDistribution and GammaDistribution.
- PearsonDistribution[4,…] is not related to standard distributions.
- PearsonDistribution[5,…] is a shifted InverseGammaDistribution.
- PearsonDistribution[6,…] is a shifted and rescaled FRatioDistribution.
- PearsonDistribution[7,…] is a shifted and rescaled StudentTDistribution.
- With symbolic parameters and no type argument, the first type whose parameter assumptions are not explicitly violated is assumed. Types are tried in the order 4, 1, 6, 3, 5, 2, and 7.
- The parameter assumptions can be obtained from DistributionParameterAssumptions.
- PearsonDistribution allows a1, a0, b2, b1, and b0 to be quantities such that one can find a unit for x that makes
dimensionless. »
- PearsonDistribution can be used with such functions as Mean, CDF, and RandomVariate.
Background & Context
- PearsonDistribution represents a statistical distribution belonging to one of seven types as determined by argument structure. Pearson distributions originate with English mathematician Karl Pearson, who devised them in order to model distributions that are visibly skewed.
- The overall shape of the probability density function (PDF) of a Pearson distribution varies significantly based on its arguments. For example, the PDF of type I Pearson distributions may be either monotonic increasing, monotonic decreasing, or may have a single "peak" (i.e. a global maximum), whereas the PDF of type IV Pearson distributions always has a single peak and looks similar to skewed, asymmetric Gaussian distributions. In addition, the PDFs of various types of PearsonDistribution may be defined and supported over different types of intervals (for example, the domain of a type I Pearson is a bounded, finite-length interval, whereas the domain of a type IV is all of ), and 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 type. (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.)
- Pearson type IV is commonly used to fit distributions obtained from data or from Monte Carlo simulations, whereas the other Pearson families are intended to approximate unimodal distributions that are modeled well by type IV but not by other more "standard" distributions. Many distributions are described by (or are limiting values and/or special cases of) families of Pearson distributions, meaning Pearson distributions are extremely general in the types of phenomena they may model. For example, certain types of Pearson distributions play fundamental roles in describing disease transmission behavior, properties of Wiener processes, fundamental concepts in Bayesian statistics, the size of insurance claims, and bacterial gene expression.
- RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a Pearson distribution. Distributed[x,PearsonDistribution[type,a1,a0,b2,b1,b0]], written more concisely as xPearsonDistribution[type,a1,a0,b2,b1,b0], can be used to assert that a random variable x is distributed according to a Pearson 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 Pearson distributions of a given type may be given using PDF[PearsonDistribution[type,a1,a0,b2,b1,b0],x] and CDF[PearsonDistribution[type,a1,a0,b2,b1,b0],x]. Pearson distributions are special in the sense that their PDF satisfies a first-order differential equation involving a simple rational function of the form
. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively. When a Pearson distribution is finite, its first four moments uniquely determine it.
- DistributionFitTest can be used to test if a given dataset is consistent with a Pearson distribution, EstimatedDistribution to estimate a Pearson parametric distribution from given data, and FindDistributionParameters to fit data to a Pearson distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic Pearson distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic Pearson distribution.
- TransformedDistribution can be used to represent a transformed Pearson 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 Pearson distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving Pearson distributions.
- PearsonDistribution is closely related to a number of other distributions. For example, types I and II Pearson distributions are shifted and rescaled versions of BetaDistribution, type III generalizes both NormalDistribution and GammaDistribution, type V is a shifted version of InverseGammaDistribution, and types VI and VII are shifted and rescaled versions of FRatioDistribution and StudentTDistribution, respectively. Though type IV Pearson distributions are unrelated to other standard distributions in this usual sense, they have PDFs that appear to be asymmetric versions of StudentTDistribution. Furthermore, for certain argument values, type IV Pearson distributions become generalizations of CauchyDistribution. PearsonDistribution is also closely related to ArcSinDistribution, BetaPrimeDistribution, PowerDistribution, ParetoDistribution, LevyDistribution, InverseChiSquareDistribution, HotellingTSquareDistribution, HalfNormalDistribution, and ErlangDistribution.
Examples
open allclose allBasic Examples (4)Summary of the most common use cases

https://wolfram.com/xid/0cf2cwmwrl3vkii-b2fu6h


https://wolfram.com/xid/0cf2cwmwrl3vkii-m9dff4


https://wolfram.com/xid/0cf2cwmwrl3vkii-3vaj5h

Cumulative distribution function:

https://wolfram.com/xid/0cf2cwmwrl3vkii-r723g


https://wolfram.com/xid/0cf2cwmwrl3vkii-j6aki4


https://wolfram.com/xid/0cf2cwmwrl3vkii-igky0d

Mean and variance of Pearson type 4:

https://wolfram.com/xid/0cf2cwmwrl3vkii-flk83h


https://wolfram.com/xid/0cf2cwmwrl3vkii-2yg9i


https://wolfram.com/xid/0cf2cwmwrl3vkii-b3jhgq

Scope (8)Survey of the scope of standard use cases
Generate a sample of pseudorandom numbers from a Pearson distribution:

https://wolfram.com/xid/0cf2cwmwrl3vkii-b7rt2v

https://wolfram.com/xid/0cf2cwmwrl3vkii-fh95i

Distribution parameters estimation:

https://wolfram.com/xid/0cf2cwmwrl3vkii-45b7g2
Estimate the distribution parameters from sample data:

https://wolfram.com/xid/0cf2cwmwrl3vkii-epi747

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

https://wolfram.com/xid/0cf2cwmwrl3vkii-f8ui5o


https://wolfram.com/xid/0cf2cwmwrl3vkii-lvfv01


https://wolfram.com/xid/0cf2cwmwrl3vkii-54wnsr


https://wolfram.com/xid/0cf2cwmwrl3vkii-ne54uk


https://wolfram.com/xid/0cf2cwmwrl3vkii-tdfjid


https://wolfram.com/xid/0cf2cwmwrl3vkii-0zlacd


https://wolfram.com/xid/0cf2cwmwrl3vkii-bpy8e9

Different moments with closed forms as functions of parameters of Pearson type 4:

https://wolfram.com/xid/0cf2cwmwrl3vkii-js043h

https://wolfram.com/xid/0cf2cwmwrl3vkii-rx074o


https://wolfram.com/xid/0cf2cwmwrl3vkii-d71a5p


https://wolfram.com/xid/0cf2cwmwrl3vkii-pknsqa


https://wolfram.com/xid/0cf2cwmwrl3vkii-6f3ti3


https://wolfram.com/xid/0cf2cwmwrl3vkii-zg9ct4


https://wolfram.com/xid/0cf2cwmwrl3vkii-9gzmth


https://wolfram.com/xid/0cf2cwmwrl3vkii-45gate


https://wolfram.com/xid/0cf2cwmwrl3vkii-rqy12o


https://wolfram.com/xid/0cf2cwmwrl3vkii-pgdppw


https://wolfram.com/xid/0cf2cwmwrl3vkii-joadzg


https://wolfram.com/xid/0cf2cwmwrl3vkii-gqkhgo


https://wolfram.com/xid/0cf2cwmwrl3vkii-z4fo9n

Consistent use of Quantity in parameters yields QuantityDistribution:

https://wolfram.com/xid/0cf2cwmwrl3vkii-hmsba2

Find skewness of the option price under this model:

https://wolfram.com/xid/0cf2cwmwrl3vkii-bo77b6

Applications (3)Sample problems that can be solved with this function
PearsonDistribution of type 4 is the only type not related to other standard distributions:

https://wolfram.com/xid/0cf2cwmwrl3vkii-bl435p


https://wolfram.com/xid/0cf2cwmwrl3vkii-d1f27


https://wolfram.com/xid/0cf2cwmwrl3vkii-cb7o03

https://wolfram.com/xid/0cf2cwmwrl3vkii-ga82zq

Find the probability for a Pearson IV random variate to be outside the plotted region:

https://wolfram.com/xid/0cf2cwmwrl3vkii-h1jazq

Moments of PearsonDistribution satisfy a three-term recurrence equation implied by the defining differential equation for the density function :

https://wolfram.com/xid/0cf2cwmwrl3vkii-bzb1sb
Express moment equations using standardized central moments:

https://wolfram.com/xid/0cf2cwmwrl3vkii-b3bp73

Augment equations to fix coefficient normalization:

https://wolfram.com/xid/0cf2cwmwrl3vkii-cjvcul

https://wolfram.com/xid/0cf2cwmwrl3vkii-d1nx52

Define Pearson distribution in terms of standardized central moments:

https://wolfram.com/xid/0cf2cwmwrl3vkii-b45n4


https://wolfram.com/xid/0cf2cwmwrl3vkii-uzepc


https://wolfram.com/xid/0cf2cwmwrl3vkii-kytsd9


https://wolfram.com/xid/0cf2cwmwrl3vkii-cmzkt9


https://wolfram.com/xid/0cf2cwmwrl3vkii-d348kk

Define Pearson distribution with zero mean and unit variance, and parameterized by skewness and kurtosis:

https://wolfram.com/xid/0cf2cwmwrl3vkii-hshyga
Obtain parameter inequalities for Pearson types 1, 4, and 6:

https://wolfram.com/xid/0cf2cwmwrl3vkii-46zj1


https://wolfram.com/xid/0cf2cwmwrl3vkii-26mr9

https://wolfram.com/xid/0cf2cwmwrl3vkii-hp3zyw
Determine the type of PearsonDistribution whose moments match sampling moments:

https://wolfram.com/xid/0cf2cwmwrl3vkii-4eqpc

Compare with the estimated distribution:

https://wolfram.com/xid/0cf2cwmwrl3vkii-qr39o

Properties & Relations (24)Properties of the function, and connections to other functions
Certain members of the PearsonDistribution family are closed under affine transforms:

https://wolfram.com/xid/0cf2cwmwrl3vkii-etmnr


https://wolfram.com/xid/0cf2cwmwrl3vkii-eqgrt2

Relationships to other distributions:

ArcSinDistribution is a special type of Pearson type 1 and type 2 distributions:

https://wolfram.com/xid/0cf2cwmwrl3vkii-gd4fsp


https://wolfram.com/xid/0cf2cwmwrl3vkii-3tja7u


https://wolfram.com/xid/0cf2cwmwrl3vkii-vh9gg0


https://wolfram.com/xid/0cf2cwmwrl3vkii-erhmc9

BetaDistribution is a special case of Pearson type 1 distribution:

https://wolfram.com/xid/0cf2cwmwrl3vkii-x3yp0y


https://wolfram.com/xid/0cf2cwmwrl3vkii-negofw


https://wolfram.com/xid/0cf2cwmwrl3vkii-nktwja

PowerDistribution is a special case of Pearson type 1 distribution:

https://wolfram.com/xid/0cf2cwmwrl3vkii-6ng28y


https://wolfram.com/xid/0cf2cwmwrl3vkii-01h3pi


https://wolfram.com/xid/0cf2cwmwrl3vkii-9itnye

WignerSemicircleDistribution is a special case of Pearson type 1 and type 2 distributions:

https://wolfram.com/xid/0cf2cwmwrl3vkii-0y5as3


https://wolfram.com/xid/0cf2cwmwrl3vkii-muqju3


https://wolfram.com/xid/0cf2cwmwrl3vkii-smoa0u


https://wolfram.com/xid/0cf2cwmwrl3vkii-ylwno8

ChiSquareDistribution is a special case of Pearson type 3 distribution:

https://wolfram.com/xid/0cf2cwmwrl3vkii-2a1tgg


https://wolfram.com/xid/0cf2cwmwrl3vkii-j3sph5


https://wolfram.com/xid/0cf2cwmwrl3vkii-1ijdyq

ErlangDistribution is a special case of Pearson type 3 distribution:

https://wolfram.com/xid/0cf2cwmwrl3vkii-4gzlwb


https://wolfram.com/xid/0cf2cwmwrl3vkii-rikkdt


https://wolfram.com/xid/0cf2cwmwrl3vkii-q1f88d

ExponentialDistribution is a special case of Pearson type 3 distribution:

https://wolfram.com/xid/0cf2cwmwrl3vkii-q9ld0b


https://wolfram.com/xid/0cf2cwmwrl3vkii-3cr68n


https://wolfram.com/xid/0cf2cwmwrl3vkii-ost8vz

GammaDistribution is a special case of Pearson type 3 distribution:

https://wolfram.com/xid/0cf2cwmwrl3vkii-yh1j16


https://wolfram.com/xid/0cf2cwmwrl3vkii-bcxz8w


https://wolfram.com/xid/0cf2cwmwrl3vkii-6o27vn

Scaled HalfNormalDistribution is a special case of Pearson type 3 distribution:

https://wolfram.com/xid/0cf2cwmwrl3vkii-yknk7q


https://wolfram.com/xid/0cf2cwmwrl3vkii-mtf1cd


https://wolfram.com/xid/0cf2cwmwrl3vkii-4hw769


https://wolfram.com/xid/0cf2cwmwrl3vkii-x6b3l1

NormalDistribution is a special case of Pearson type 3 distribution:

https://wolfram.com/xid/0cf2cwmwrl3vkii-xnrrmj


https://wolfram.com/xid/0cf2cwmwrl3vkii-j864on


https://wolfram.com/xid/0cf2cwmwrl3vkii-dh0y64

CauchyDistribution is a limiting case of Pearson type 4 distribution:

https://wolfram.com/xid/0cf2cwmwrl3vkii-lbuz5v


https://wolfram.com/xid/0cf2cwmwrl3vkii-pk8b05


https://wolfram.com/xid/0cf2cwmwrl3vkii-rr75s9

CauchyDistribution is a special case of Pearson type 7 distribution:

https://wolfram.com/xid/0cf2cwmwrl3vkii-yioikr


https://wolfram.com/xid/0cf2cwmwrl3vkii-k6ca4y


https://wolfram.com/xid/0cf2cwmwrl3vkii-rguy6p

StudentTDistribution is a special case of Pearson type 4 and type 7 distributions:

https://wolfram.com/xid/0cf2cwmwrl3vkii-nskf6v


https://wolfram.com/xid/0cf2cwmwrl3vkii-qvx0i8


https://wolfram.com/xid/0cf2cwmwrl3vkii-c9smn1


https://wolfram.com/xid/0cf2cwmwrl3vkii-0981af

Generalized StudentTDistribution is a special case of Pearson type 4 and type 7 distributions:

https://wolfram.com/xid/0cf2cwmwrl3vkii-cvm47m


https://wolfram.com/xid/0cf2cwmwrl3vkii-mlp0za


https://wolfram.com/xid/0cf2cwmwrl3vkii-6acro0


https://wolfram.com/xid/0cf2cwmwrl3vkii-m5bmdb

InverseChiSquareDistribution is a special case of Pearson type 5 distribution:

https://wolfram.com/xid/0cf2cwmwrl3vkii-uaqgc7


https://wolfram.com/xid/0cf2cwmwrl3vkii-b457ad


https://wolfram.com/xid/0cf2cwmwrl3vkii-yfws20

Scaled InverseChiSquareDistribution is a special case of Pearson type 5 distribution:

https://wolfram.com/xid/0cf2cwmwrl3vkii-ovb393


https://wolfram.com/xid/0cf2cwmwrl3vkii-o6um18


https://wolfram.com/xid/0cf2cwmwrl3vkii-nobrgg

InverseGammaDistribution is a special case of Pearson type 5 distribution:

https://wolfram.com/xid/0cf2cwmwrl3vkii-9hzd6o


https://wolfram.com/xid/0cf2cwmwrl3vkii-84oc9z


https://wolfram.com/xid/0cf2cwmwrl3vkii-ysgv6j

LevyDistribution is a special case of Pearson type 5 distribution:

https://wolfram.com/xid/0cf2cwmwrl3vkii-3ue547


https://wolfram.com/xid/0cf2cwmwrl3vkii-b2jca6


https://wolfram.com/xid/0cf2cwmwrl3vkii-84pwjv

BetaPrimeDistribution is a special case of Pearson type 6 distribution:

https://wolfram.com/xid/0cf2cwmwrl3vkii-wtirf0


https://wolfram.com/xid/0cf2cwmwrl3vkii-6w21qe


https://wolfram.com/xid/0cf2cwmwrl3vkii-m78srd

FRatioDistribution is a special case of Pearson type 6 distribution:

https://wolfram.com/xid/0cf2cwmwrl3vkii-5blvj7


https://wolfram.com/xid/0cf2cwmwrl3vkii-xyd3pk


https://wolfram.com/xid/0cf2cwmwrl3vkii-wgyuig

HotellingTSquareDistribution is a special case of Pearson type 6 distribution:

https://wolfram.com/xid/0cf2cwmwrl3vkii-skz0uo


https://wolfram.com/xid/0cf2cwmwrl3vkii-exw3x3


https://wolfram.com/xid/0cf2cwmwrl3vkii-whoxzk

ParetoDistribution is a special case of Pearson type 6 distribution:

https://wolfram.com/xid/0cf2cwmwrl3vkii-prarfb


https://wolfram.com/xid/0cf2cwmwrl3vkii-es1yyr


https://wolfram.com/xid/0cf2cwmwrl3vkii-ylnd93

Wolfram Research (2010), PearsonDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/PearsonDistribution.html (updated 2016).
Text
Wolfram Research (2010), PearsonDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/PearsonDistribution.html (updated 2016).
Wolfram Research (2010), PearsonDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/PearsonDistribution.html (updated 2016).
CMS
Wolfram Language. 2010. "PearsonDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/PearsonDistribution.html.
Wolfram Language. 2010. "PearsonDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/PearsonDistribution.html.
APA
Wolfram Language. (2010). PearsonDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PearsonDistribution.html
Wolfram Language. (2010). PearsonDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PearsonDistribution.html
BibTeX
@misc{reference.wolfram_2025_pearsondistribution, author="Wolfram Research", title="{PearsonDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/PearsonDistribution.html}", note=[Accessed: 10-July-2025
]}
BibLaTeX
@online{reference.wolfram_2025_pearsondistribution, organization={Wolfram Research}, title={PearsonDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/PearsonDistribution.html}, note=[Accessed: 10-July-2025
]}