# TracyWidomDistribution

represents a TracyWidom distribution with Dyson index β.

# Details • The TracyWidom distribution is the limiting probability distribution of the normalized largest eigenvalue of a random matrix that belongs to a Gaussian ensemble.
• TracyWidomDistribution allows the Dyson index β to be 1, 2, or 4 according to the underlying matrix distribution:
• GaussianOrthogonalMatrixDistribution β2 GaussianUnitaryMatrixDistribution β4 GaussianSymplecticMatrixDistribution
• TracyWidomDistribution 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:

## Scope(4)

Generate a sample of pseudorandom numbers from a TracyWidom distribution:

Compare its histogram to the PDF:

Moments of TracyWidom distributions are not available in closed form:

Find their machineprecision approximation:

Compute the Mean of TracyWidom distribution to 50digit precision:

Hazard function:

Quantile function:

## Applications(4)

Use MatrixPropertyDistribution to represent the normalized largest eigenvalue of a matrix from Gaussian unitary ensemble:

Sample from MatrixPropertyDistribution:

Compare the histogram with the PDF:

Perform similar computations with Gaussian orthogonal ensemble and Gaussian symplectic ensemble:

Compare the histograms with the PDFs:

When n and p (the dimension of the covariance matrix Σ) are both large, the scaled largest eigenvalue of a matrix from Wishart ensemble with identity covariance is approximately distributed as TracyWidom distribution of :

Sample the scaled largest eigenvalue:

Compare the histogram of scaled largest eigenvalues with the PDF:

Compare the CDFs of TracyWidom distributions with the leading order asymptotic expansion on the left tail in log scale:

Define a function to find the length of LongestOrderedSequence in the given sequence:

Find lengths of the longest increasing subsequence in random permutations of list :

Compare the scaled length of the longest increasing subsequence with TracyWidom distribution of :

## Properties & Relations(2)

CDFs of TracyWidom distributions for different values of β are related to each other:

TracyWidom distributions can be well approximated by GammaDistribution in the central region:

Match GammaDistribution with TracyWidom distribution of with the first three moments:

Compare the PDFs between :

Compare the CDFs between in log scale: