Multivariate Normal Distribution
A p-variate multivariate normal distribution (also called a multinormal distribution) is a generalization of the bivariate normal distribution. The p-multivariate distribution with mean vector mu and covariance matrix Sigma is denoted N_p(mu,Sigma). The multivariate normal distribution is implemented as MultinormalDistribution [{mu1, mu2, ...}, {{sigma11, sigma12, ...}, {sigma12, sigma22, ..., }...}, {x1, x2, ...}] in the Wolfram Language package MultivariateStatistics` (where the matrix Sigma must be symmetric since sigma_(ij)=sigma_(ji)).
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.
See also
Bivariate Normal Distribution, Gaussian Joint Variable Theorem, Normal Distribution, Trivariate Normal DistributionExplore with Wolfram|Alpha
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.Referenced on Wolfram|Alpha
Multivariate Normal DistributionCite this as:
Weisstein, Eric W. "Multivariate Normal Distribution." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MultivariateNormalDistribution.html