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Multivariate normal distribution

In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be k-variate normally distributed if every linear combination of…

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CF
exp ( i μ T t − 1 2 t T Σ t ) {\displaystyle \exp \!{\Big (}i{\boldsymbol {\mu }}^{\mathrm {T} }\mathbf {t} -{\tfrac {1}{2}}\mathbf {t} ^{\mathrm {T} }{\boldsymbol {\Sigma }}\ma…
Entropy
k 2 log ⁡ ( 2 π e ) + 1 2 log ⁡ det ( Σ ) {\displaystyle {\frac {k}{2}}\log {\mathord {\left(2\pi \mathrm {e} \right)}}+{\frac {1}{2}}\log \det {\mathord {\left({\boldsymbol {\S…
Kullback–Leibler divergence
See § Kullback–Leibler divergence
Mean
μ
MGF
exp ( μ T t + 1 2 t T Σ t ) {\displaystyle \exp \!{\Big (}{\boldsymbol {\mu }}^{\mathrm {T} }\mathbf {t} +{\tfrac {1}{2}}\mathbf {t} ^{\mathrm {T} }{\boldsymbol {\Sigma }}\mathb…
Mode
μ

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Multivariate normal distribution

Nodes114
Edges113
Triples86
Avg. degree1.98
Density0.017544
Components1

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Multivariate normal distribution

Top relations

related to Drawing values from the distribution · 14
Multivariate normal distribution → AAT, Az, Box, Cholesky, Find, If, LAPACK's, Let, Muller, N-dimensional, This, UΛ1/2, UΛU, When
related to Affine transformation · 11
Multivariate normal distribution → BX, BΣBT, Bμ, If, In, Sigma, To, X1, X2, X4, Xi
related to Multivariate normality tests · 10
Multivariate normal distribution → Cox, Friedman, Jain's, Jerome Friedman, Larry Rafsky, Multivariate, Rafsky, Small, Smith, The
see also · 9
Multivariate normal distribution → Chi, Complex, Euclidean, Gaussian, Hoyt, Mahalanobis, Multivariate, Rayleigh, Rice
related to Equivalent definitions · 6
Multivariate normal distribution → Big, Every, Sigma, That, The, There
related to Geometric interpretation · 6
Multivariate normal distribution → Hence, If, Sigma, The, UΛ1/2, UΛUT
related to Bayesian inference · 3
Multivariate normal distribution → In Bayesian, Suppose, Wishart
related to Marginal distributions · 3
Multivariate normal distribution → Example, The, To
related to Mutual information · 3
Multivariate normal distribution → Kullback, Leibler, The
is a · 2
Multivariate normal distribution → example of the class of elliptical distributions, special case of the Kullback

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Important terminology

displaystyle distribution normal multivariate boldsymbol matrix sigma vector mathbf covariance random mu case mean density function variables independent two probability

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Multivariate normal distributionCFexp ( i μ T t − 1 2 t T Σ t ) {\displaystyle \exp \!{\Big (}i{\boldsymbol {\mu }}^{\mathrm {T} }\mathbf {t} -{\tfrac {1}{2}}\mathbf {t} ^{\mathrm {T} }{\boldsymbol {\Sigma }}\ma…1.00infobox
Multivariate normal distributionEntropyk 2 log ⁡ ( 2 π e ) + 1 2 log ⁡ det ( Σ ) {\displaystyle {\frac {k}{2}}\log {\mathord {\left(2\pi \mathrm {e} \right)}}+{\frac {1}{2}}\log \det {\mathord {\left({\boldsymbol {\S…1.00infobox
Multivariate normal distributionKullback–Leibler divergenceSee § Kullback–Leibler divergence1.00infobox
Multivariate normal distributionMeanμ1.00infobox
Multivariate normal distributionMGFexp ( μ T t + 1 2 t T Σ t ) {\displaystyle \exp \!{\Big (}{\boldsymbol {\mu }}^{\mathrm {T} }\mathbf {t} +{\tfrac {1}{2}}\mathbf {t} ^{\mathrm {T} }{\boldsymbol {\Sigma }}\mathb…1.00infobox
Multivariate normal distributionModeμ1.00infobox
Multivariate normal distributionNotationN ( μ , Σ ) {\displaystyle {\mathcal {N}}({\boldsymbol {\mu }},\,{\boldsymbol {\Sigma }})}1.00infobox
Multivariate normal distributionParametersμ ∈ Rk — location Σ ∈ Rk × k — covariance (positive semi-definite matrix)1.00infobox
Multivariate normal distributionPDF( 2 π ) − k / 2 det ( Σ ) − 1 / 2 exp ⁡ ( − 1 2 ( x − μ ) T Σ − 1 ( x − μ ) ) , {\displaystyle (2\pi )^{-k/2}\det({\boldsymbol {\Sigma }})^{-1/2}\,\exp \left(-{\frac {1}{2}}(\ma…1.00infobox
Multivariate normal distributionSupportx ∈ μ + span(Σ) ⊆ Rk1.00infobox
Multivariate normal distributionVarianceΣ, the matrix of individual variances and covariances1.00infobox
Multivariate normal distributionis aspecial case of the Kullback0.90text
Multivariate normal distributionis aexample of the class of elliptical distributions0.90text

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