WOLFRAM

represents a TracyWidom distribution with Dyson index β.

Details

Examples

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Basic Examples  (4)Summary of the most common use cases

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  (4)Survey of the scope of standard use cases

Generate a sample of pseudorandom numbers from a TracyWidom distribution:

Compare its histogram to the PDF:

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Moments of TracyWidom distributions are not available in closed form:

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Find their machineprecision approximation:

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Compute the Mean of TracyWidom distribution to 50digit precision:

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Hazard function:

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Quantile function:

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Applications  (4)Sample problems that can be solved with this function

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

Sample from MatrixPropertyDistribution:

Compare the histogram with the PDF:

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

Compare the histograms with the PDFs:

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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:

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

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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 :

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Properties & Relations  (2)Properties of the function, and connections to other functions

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

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

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

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Compare the PDFs between :

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Compare the CDFs between in log scale:

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Wolfram Research (2015), TracyWidomDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/TracyWidomDistribution.html.
Wolfram Research (2015), TracyWidomDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/TracyWidomDistribution.html.

Text

Wolfram Research (2015), TracyWidomDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/TracyWidomDistribution.html.

Wolfram Research (2015), TracyWidomDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/TracyWidomDistribution.html.

CMS

Wolfram Language. 2015. "TracyWidomDistribution." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/TracyWidomDistribution.html.

Wolfram Language. 2015. "TracyWidomDistribution." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/TracyWidomDistribution.html.

APA

Wolfram Language. (2015). TracyWidomDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/TracyWidomDistribution.html

Wolfram Language. (2015). TracyWidomDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/TracyWidomDistribution.html

BibTeX

@misc{reference.wolfram_2025_tracywidomdistribution, author="Wolfram Research", title="{TracyWidomDistribution}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/TracyWidomDistribution.html}", note=[Accessed: 29-March-2025 ]}

@misc{reference.wolfram_2025_tracywidomdistribution, author="Wolfram Research", title="{TracyWidomDistribution}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/TracyWidomDistribution.html}", note=[Accessed: 29-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_tracywidomdistribution, organization={Wolfram Research}, title={TracyWidomDistribution}, year={2015}, url={https://reference.wolfram.com/language/ref/TracyWidomDistribution.html}, note=[Accessed: 29-March-2025 ]}

@online{reference.wolfram_2025_tracywidomdistribution, organization={Wolfram Research}, title={TracyWidomDistribution}, year={2015}, url={https://reference.wolfram.com/language/ref/TracyWidomDistribution.html}, note=[Accessed: 29-March-2025 ]}