Probability density function:

Cumulative distribution function:

Mean and variance:

Generate a sample of pseudorandom numbers from a Marchenko–Pastur distribution with :

Compare its histogram to the PDF:

Generate a sample of pseudorandom numbers from a Marchenko–Pastur 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:

Moment:

Closed form for symbolic order:

CentralMoment:

Closed form for symbolic order:

FactorialMoment:

Cumulant:

Hazard function:

Quantile function:

Consistent use of Quantity in parameters yields QuantityDistribution:

Find the median area:

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

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

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:

Marchenko–Pastur distribution with is a mixed type distribution, which is neither continuous nor discrete:

The CDF for such Marchenko–Pastur distributions is discontinuous at :

The probability density function for Marchenko–Pastur 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: