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

多変量正規ベクトル z の二次形式 z.a.z+b.z+c の分布を表す．

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例題  
例  
スコープ  
考えられる問題  
テクニカルノート
関連するガイド
引用書式
MultivariateStatistics`
MultivariateStatistics`

QuadraticFormDistribution

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

多変量正規ベクトル z の二次形式 z.a.z+b.z+c の分布を表す．

詳細とオプション

例題

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例  (2)

Wolfram Language code: Needs["MultivariateStatistics`"]

二次形式分布の平均：

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`"]

二次形式分布の分散：

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}}}]]

スコープ  (3)

Wolfram Language code: Needs["MultivariateStatistics`"]

二次形式分布に従う擬似乱数集合を発生させる：

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}}}]]

考えられる問題  (2)

Wolfram Language code: Needs["MultivariateStatistics`"]

確率密度関数と累積分布関数は，Seriesを使わないと評価できない：

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}]

級数展開は分布の下部サポート点付近でなければならない：

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

無効なパラメータを記号的出力に代入すると，意味をなさない結果となる：

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言語関数， https://reference.wolfram.com/language/MultivariateStatistics/ref/QuadraticFormDistribution.html.

テキスト

Wolfram Research (2007), QuadraticFormDistribution, Wolfram言語関数， 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: 18-August-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: 18-August-2026]}