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_2024_pearsondistribution, author="Wolfram Research", title="{PearsonDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/PearsonDistribution.html}", note=[Accessed: 10-January-2025
]}
BibLaTeX
@online{reference.wolfram_2024_pearsondistribution, organization={Wolfram Research}, title={PearsonDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/PearsonDistribution.html}, note=[Accessed: 10-January-2025
]}