BUILT-IN MATHEMATICA SYMBOL

WignerSemicircleDistribution

WignerSemicircleDistribution[r]
represents a Wigner semicircle distribution with radius r centered at the origin.

represents a Wigner semicircle distribution with radius r centered at a.

MORE INFORMATION

The probability density for value in a Wigner semicircle distribution is proportional to for between and .

WignerSemicircleDistribution allows a to be any real number and r to be any positive real number.

WignerSemicircleDistribution can be used with such functions as Mean, CDF, and RandomVariate.

EXAMPLES

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Basic Examples (4)

Probability density function:

Cumulative distribution function:

Mean and variance:

Median:

Probability density function:

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Cumulative distribution function:

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Mean and variance:

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Median:

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Scope (6)

Generate a set of pseudorandom numbers that have Wigner semicircle distribution:

Compare its histogram to the PDF:

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 and kurtosis are constant:

Different moments with closed forms as functions of parameters:

Moment:

Closed form for symbolic order:

CentralMoment:

Closed form for symbolic order:

FactorialMoment:

Cumulant:

Hazard function:

Quantile function:

Applications (1)

Construct a Gaussian symmetric matrix:

Fit WignerSemicircleDistribution into the eigenvalues:

Compare the histogram of the eigenvalues with the PDF:

Properties & Relations (4)

Parameter influence on the CDF for each

:

The WignerSemicircleDistribution is closed under translation and scaling by a positive factor:

Relationships to other distributions:

Wigner semicircle distribution is a special case of type 1 PearsonDistribution: