# PearsonDistribution

PearsonDistribution[a1,a0,b2,b1,b0]

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

PearsonDistribution[type,a1,a0,b2,b1,b0]

represents a Pearson distribution of given type.

# Details # 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

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## Basic Examples(4)

Probability density function:

Cumulative distribution function:

Mean and variance of Pearson type 4:

Pearson type 5 distribution:

## Scope(8)

Generate a sample of pseudorandom numbers from a Pearson distribution:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

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

Skewness:

Kurtosis:

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

Hazard function:

Quantile function:

Consistent use of Quantity in parameters yields QuantityDistribution:

Find skewness of the option price under this model:

## Applications(3)

PearsonDistribution of type 4 is the only type not related to other standard distributions:

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

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

Express moment equations using standardized central moments:

Augment equations to fix coefficient normalization:

Solve equations:

Define Pearson distribution in terms of standardized central moments:

Check the solution:

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

Obtain parameter inequalities for Pearson types 1, 4, and 6:

Determine the type of PearsonDistribution whose moments match sampling moments:

Compare with the estimated distribution:

## Properties & Relations(24)

Certain members of the PearsonDistribution family are closed under affine transforms:

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

BetaDistribution is a special case of Pearson type 1 distribution:

PowerDistribution is a special case of Pearson type 1 distribution:

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

ChiSquareDistribution is a special case of Pearson type 3 distribution:

ErlangDistribution is a special case of Pearson type 3 distribution:

ExponentialDistribution is a special case of Pearson type 3 distribution:

GammaDistribution is a special case of Pearson type 3 distribution:

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

NormalDistribution is a special case of Pearson type 3 distribution:

CauchyDistribution is a limiting case of Pearson type 4 distribution:

CauchyDistribution is a special case of Pearson type 7 distribution:

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

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

InverseChiSquareDistribution is a special case of Pearson type 5 distribution:

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

InverseGammaDistribution is a special case of Pearson type 5 distribution:

LevyDistribution is a special case of Pearson type 5 distribution:

BetaPrimeDistribution is a special case of Pearson type 6 distribution:

FRatioDistribution is a special case of Pearson type 6 distribution:

HotellingTSquareDistribution is a special case of Pearson type 6 distribution:

ParetoDistribution is a special case of Pearson type 6 distribution:

## Neat Examples(1)

PDFs for different values with CDF contours: