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HotellingTSquareDistribution[p,m]

represents Hotelling's distribution with dimensionality parameter p and degrees of freedom parameter m.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Properties & Relations  
Possible Issues  
See Also
Tech Notes
Related Guides
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MultivariateStatistics`
MultivariateStatistics`

HotellingTSquareDistribution

As of Version 8, HotellingTSquareDistribution is part of the built-in Wolfram Language kernel.

HotellingTSquareDistribution[p,m]

represents Hotelling's distribution with dimensionality parameter p and degrees of freedom parameter m.

Details and Options

  • To use HotellingTSquareDistribution, you first need to load the Multivariate Statistics Package using Needs["MultivariateStatistics`"].
  • HotellingTSquareDistribution is a univariate distribution derived from the multivariate normal distribution.
  • The probability density for value x in Hotelling's distribution is proportional to x-1+p/2(1+x/m)-(m+1)/2 for x>0.
  • The parameters p and m can be any positive real numbers such that m>p-1.
  • HotellingTSquareDistribution can be used with such functions as Mean, CDF, and RandomReal.

Examples

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Basic Examples  (3)

Wolfram Language code: Needs["MultivariateStatistics`"]

The mean of a Hotelling distribution:

Wolfram Language code: Mean[HotellingTSquareDistribution[p, m]]
Wolfram Language code: Needs["MultivariateStatistics`"]

The variance:

Wolfram Language code: Variance[HotellingTSquareDistribution[p, m]]
Wolfram Language code: Needs["MultivariateStatistics`"]

Probability density function:

Wolfram Language code: PDF[HotellingTSquareDistribution[p, m], x]
Wolfram Language code: Plot[PDF[HotellingTSquareDistribution[5, 10], x], {x, 0, 50}]

Scope  (3)

Wolfram Language code: Needs["MultivariateStatistics`"]

Generate a set of pseudorandom numbers that follow a Hotelling distribution:

Wolfram Language code: RandomReal[HotellingTSquareDistribution[5, 10], 10]
Wolfram Language code: Needs["MultivariateStatistics`"]
Wolfram Language code: Skewness[HotellingTSquareDistribution[p, m]]
Wolfram Language code: Needs["MultivariateStatistics`"]
Wolfram Language code: Kurtosis[HotellingTSquareDistribution[p, m]]

Properties & Relations  (1)

Wolfram Language code: Needs["MultivariateStatistics`"]

The probability density function integrates to unity:

Wolfram Language code: Integrate[PDF[HotellingTSquareDistribution[p, m], x], {x, 0, ∞}, Assumptions -> 0 < p < m]

Hotelling variables are related to F-ratio variables by a multiplicative constant:

Wolfram Language code: y = x(m p / (m - p + 1)); PDF[HotellingTSquareDistribution[p, m], y] * D[y, x]
Wolfram Language code: PDF[FRatioDistribution[p, 1 + m - p], x]
Wolfram Language code: FullSimplify[% == %, Assumptions -> {x > 0, m > p > 0}]

Possible Issues  (2)

Wolfram Language code: Needs["MultivariateStatistics`"]

HotellingTSquareDistribution is not defined when m<p:

Wolfram Language code: Mean[HotellingTSquareDistribution[5, 2]]

HotellingTSquareDistribution is not defined when p is less than 0:

Wolfram Language code: Mean[HotellingTSquareDistribution[-5, 2]]
Wolfram Language code: Needs["MultivariateStatistics`"]

Substitution of invalid parameters into symbolic outputs gives results that are not meaningful:

Wolfram Language code: Mean[HotellingTSquareDistribution[p, m]] /. {p -> 1, m -> I}
Wolfram Research (2007), HotellingTSquareDistribution, Wolfram Language function, https://reference.wolfram.com/language/MultivariateStatistics/ref/HotellingTSquareDistribution.html.

Text

Wolfram Research (2007), HotellingTSquareDistribution, Wolfram Language function, https://reference.wolfram.com/language/MultivariateStatistics/ref/HotellingTSquareDistribution.html.

CMS

Wolfram Language. 2007. "HotellingTSquareDistribution." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/MultivariateStatistics/ref/HotellingTSquareDistribution.html.

APA

Wolfram Language. (2007). HotellingTSquareDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/MultivariateStatistics/ref/HotellingTSquareDistribution.html

BibTeX

@misc{reference.wolfram_2026_hotellingtsquaredistribution, author="Wolfram Research", title="{HotellingTSquareDistribution}", year="2007", howpublished="\url{https://reference.wolfram.com/language/MultivariateStatistics/ref/HotellingTSquareDistribution.html}", note=[Accessed: 15-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_hotellingtsquaredistribution, organization={Wolfram Research}, title={HotellingTSquareDistribution}, year={2007}, url={https://reference.wolfram.com/language/MultivariateStatistics/ref/HotellingTSquareDistribution.html}, note=[Accessed: 15-June-2026]}