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QuadraticFormDistribution[{a,b,c},{μ,Σ}]

represents the distribution of a quadratic form z.a.z+b.z+c for multivariate normal z.

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MultivariateStatistics`
MultivariateStatistics`

QuadraticFormDistribution

QuadraticFormDistribution[{a,b,c},{μ,Σ}]

represents the distribution of a quadratic form z.a.z+b.z+c for multivariate normal z.

Details and Options

Examples

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Basic Examples  (2)

Wolfram Language code: Needs["MultivariateStatistics`"]

The mean of a quadratic form distribution:

Wolfram Language code: Mean[QuadraticFormDistribution[{{{Subscript[a, 11], Subscript[a, 12]}, {Subscript[a, 12], Subscript[a, 22]}}, {Subscript[b, 1], Subscript[b, 2]}, c}, {{Subscript[μ, 1], Subscript[μ, 2]}, {{Subscript[σ, 11]^2, ρ * Subscript[σ, 11] * Subscript[σ, 22]}, {ρ * Subscript[σ, 11] * Subscript[σ, 22], Subscript[σ, 22]^2}}}]]
Wolfram Language code: Needs["MultivariateStatistics`"]

The variance of a quadratic form distribution:

Wolfram Language code: Variance[QuadraticFormDistribution[{{{1, 1 / 2}, {1 / 2, 1}}, {Subscript[b, 1], Subscript[b, 2]}, c}, {{Subscript[μ, 1], Subscript[μ, 2]}, {{1, 0}, {0, 1}}}]]

Scope  (3)

Wolfram Language code: Needs["MultivariateStatistics`"]

Generate a set of pseudorandom numbers that follow a quadratic form distribution:

Wolfram Language code: RandomReal[QuadraticFormDistribution[{{{1, 1 / 2}, {1 / 2, 1}}, {1, 2}, 5}, {{0, 0}, {{1, 0}, {0, 1}}}], 10]
Wolfram Language code: Needs["MultivariateStatistics`"]
Wolfram Language code: Skewness[QuadraticFormDistribution[{{{1, 1 / 2}, {1 / 2, 1}}, {Subscript[b, 1], Subscript[b, 2]}, c}, {{Subscript[μ, 1], Subscript[μ, 2]}, {{1, 0}, {0, 1}}}]]
Wolfram Language code: Needs["MultivariateStatistics`"]
Wolfram Language code: Kurtosis[QuadraticFormDistribution[{{{1, 1 / 2}, {1 / 2, 1}}, {Subscript[b, 1], Subscript[b, 2]}, c}, {{Subscript[μ, 1], Subscript[μ, 2]}, {{1, 0}, {0, 1}}}]]

Possible Issues  (2)

Wolfram Language code: Needs["MultivariateStatistics`"]

PDF and CDF can only be evaluated using Series:

Wolfram Language code: dist = QuadraticFormDistribution[{{{1, 0}, {0, 1}}, {1, 2}, 0}, {{0, 0}, {{1, 0}, {0, 1}}}]
Wolfram Language code: pdf = PDF[dist, x]
Wolfram Language code: Series[pdf, {x, -5 / 4, 2}]
Wolfram Language code: Series[CDF[dist, x], {x, -5 / 4, 2}]

Series expansion must be about the lower support point for the distribution:

Wolfram Language code: Series[CDF[dist, x], {x, 0, 2}]
Wolfram Language code: Needs["MultivariateStatistics`"]

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

Wolfram Language code: Mean[QuadraticFormDistribution[{{{1, 1 / 2}, {1 / 2, 1}}, {Subscript[b, 1], Subscript[b, 2]}, c}, {{0, 1}, {{1, 0}, {0, 1}}}]] /. {Subscript[b, 1] -> 1, Subscript[b, 2] -> I, c -> I}
Wolfram Research (2007), QuadraticFormDistribution, Wolfram Language function, https://reference.wolfram.com/language/MultivariateStatistics/ref/QuadraticFormDistribution.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_quadraticformdistribution, author="Wolfram Research", title="{QuadraticFormDistribution}", year="2007", howpublished="\url{https://reference.wolfram.com/language/MultivariateStatistics/ref/QuadraticFormDistribution.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_quadraticformdistribution, organization={Wolfram Research}, title={QuadraticFormDistribution}, year={2007}, url={https://reference.wolfram.com/language/MultivariateStatistics/ref/QuadraticFormDistribution.html}, note=[Accessed: 13-June-2026]}