Multivariate Normal Distribution
المؤلف:
Rose, C. and Smith, M. D.
المصدر:
"The Multivariate Normal Distribution." Mathematica J. 6
الجزء والصفحة:
...
2-3-2021
3358
Multivariate Normal Distribution
A 
-variate multivariate normal distribution (also called a multinormal distribution) is a generalization of the bivariate normal distribution. The 
-multivariate distribution with mean vector 
and covariance matrix 
is denoted 
. The multivariate normal distribution is implemented as MultinormalDistribution[![<span style=]()
{" src="https://mathworld.wolfram.com/images/equations/MultivariateNormalDistribution/Inline6.gif" style="height:15px; width:5px" />mu1, mu2, ...![<span style=]()
}" src="https://mathworld.wolfram.com/images/equations/MultivariateNormalDistribution/Inline7.gif" style="height:15px; width:5px" />, ![<span style=]()
{" src="https://mathworld.wolfram.com/images/equations/MultivariateNormalDistribution/Inline8.gif" style="height:15px; width:5px" />![<span style=]()
{" src="https://mathworld.wolfram.com/images/equations/MultivariateNormalDistribution/Inline9.gif" style="height:15px; width:5px" />sigma11, sigma12, ...![<span style=]()
}" src="https://mathworld.wolfram.com/images/equations/MultivariateNormalDistribution/Inline10.gif" style="height:15px; width:5px" />, ![<span style=]()
{" src="https://mathworld.wolfram.com/images/equations/MultivariateNormalDistribution/Inline11.gif" style="height:15px; width:5px" />sigma12, sigma22, ..., ![<span style=]()
}" src="https://mathworld.wolfram.com/images/equations/MultivariateNormalDistribution/Inline12.gif" style="height:15px; width:5px" />...![<span style=]()
}" src="https://mathworld.wolfram.com/images/equations/MultivariateNormalDistribution/Inline13.gif" style="height:15px; width:5px" />, ![<span style=]()
{" src="https://mathworld.wolfram.com/images/equations/MultivariateNormalDistribution/Inline14.gif" style="height:15px; width:5px" />x1, x2, ...![<span style=]()
}" src="https://mathworld.wolfram.com/images/equations/MultivariateNormalDistribution/Inline15.gif" style="height:15px; width:5px" />] in the Wolfram Language package MultivariateStatistics` (where the matrix 
must be symmetric since 
).
In the case of nonzero correlations, there is in general no closed-form solution for the distribution function of a multivariate normal distribution. As a result, such computations must be done numerically.
REFERENCES:
Rose, C. and Smith, M. D. "The Multivariate Normal Distribution." Mathematica J. 6, 32-37, 1996.
Rose, C. and Smith, M. D. "Random[Title]: Manipulating Probability Density Functions." Ch. 16 in Computational Economics and Finance: Modeling and Analysis with Mathematica (Ed. H. Varian). New York: Springer-Verlag, 1996.
Rose, C. and Smith, M. D. "The Multivariate Normal Distribution." §6.4 in Mathematical Statistics with Mathematica. New York: Springer-Verlag, pp. 216-235, 2002.
Schervish, M. J. "Multivariate Normal Probabilities with Error Bounds." Appl. Stat.: J. Roy. Stat. Soc., Ser. C 33, 81-94, 1984.
Schervish, M. J. "Corrections to Multivariate Normal Probabilities with Error Bounds." Appl. Stat.: J. Roy. Stat. Soc., Ser. C 34, 103-104, 1984.
Tong, L. The Multivariate Normal Distribution. New York: Springer-Verlag, 1990.
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