This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# PearsonDistribution

 PearsonDistribution represents a distribution of the Pearson family with parameters , , , , and . PearsonDistributionrepresents a Pearson distribution of given type.
• The probability density satisfies the differential equation .
• The Pearson family of distribution is historically divided into seven types. By giving the form PearsonDistribution, the type will implicitly provide domain and parameter constraints.
• 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.
Probability density function:
Cumulative distribution function:
Mean and variance of Pearson type 4:
Pearson type 5 distribution:
Probability density function:
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Cumulative distribution function:
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Mean and variance of Pearson type 4:
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Pearson type 5 distribution:
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 Scope   (7)
Generate pseudorandom numbers that are Pearson distributed:
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:
 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:
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:
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