# MarchenkoPasturDistribution

represents a MarchenkoPastur distribution with asymptotic ratio and scale parameter .

represents a MarchenkoPastur distribution with unit scale parameter.

# Details • MarchenkoPasturDistribution is the limiting spectral density of random matrices from WishartMatrixDistribution.
• The derivative of cumulative distribution function at in a MarchenkoPastur distribution is proportional to with for between and .
• MarchenkoPastur distribution has a point mass at with probability when .
• MarchenkoPasturDistribution allows and to be any positive real numbers.
• MarchenkoPasturDistribution allows σ to be a quantity of any unit dimension, and λ to be a dimensionless quantity. »
• MarchenkoPasturDistribution can be used with such functions as Mean, CDF, and RandomVariate.

# Examples

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## Basic Examples(3)

Probability density function:

Cumulative distribution function:

Mean and variance:

## Scope(7)

Generate a sample of pseudorandom numbers from a MarchenkoPastur distribution with :

Compare its histogram to the PDF:

Generate a sample of pseudorandom numbers from a MarchenkoPastur distribution with :

Compare its cumulative histogram to the CDF:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare the cumulative histogram of the sample with the CDF of the estimated distribution:

Skewness and kurtosis depend only on :

Different moments with closed forms as functions of parameters:

Closed form for symbolic order:

Closed form for symbolic order:

Hazard function:

Quantile function:

Consistent use of Quantity in parameters yields QuantityDistribution:

Find the median area:

## Applications(1)

Use MatrixPropertyDistribution to represent the eigenvalues of a Wishart random matrix with identity covariance:

The spectral density converges to the pdf of in the limit of large and with the finite ratio :

## Properties & Relations(3)

MarchenkoPasturDistribution is closed under scaling by a positive factor:

MarchenkoPasturDistribution has an atomic weight at 0 when :

MarchenkoPasturDistribution is the limiting distribution of eigenvalues of Wishart matrices. The atomic weight at occurs when the Wishart matrix is singular. Generate a singular Wishart matrix with identity covariance and compute the scaled eigenvalues:

Fit MarchenkoPasturDistribution to the eigenvalues:

Compare the cumulative histogram of the eigenvalues with the CDF:

## Possible Issues(1)

MarchenkoPastur distribution with is a mixed type distribution, which is neither continuous nor discrete:

The CDF for such MarchenkoPastur distributions is discontinuous at :

The probability density function for MarchenkoPastur distribution with is not defined, and PDF returns unevaluated:

Differentiation of the CDF results in a function that does not integrate to one:

Computations with mixed type distributions are fully supported. Compute special moments:

Estimate parameters of Marchenko-Pastur distribution:

Introduced in 2015
(10.3)
|
Updated in 2016
(10.4)