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joint normal distribution
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(Definition)
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A finite set of random variables
 are said to have a joint normal distribution or multivariate normal distribution if their joint probability density function is multidimensional Gaussian, that is:
where
When
 is a  -dimensional random vector that is multivariate normally distributed with mean vector
 and covariance matrix
 , we often write
Like a random variable with a normal distribution, a finite set of random variables (or a random vector) with a joint normal distribution has some simple and attractive properties:
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![$ \operatorname{E}[\boldsymbol{X}]=\boldsymbol{\mu}$](http://images.planetmath.org:8080/cache/objects/7204/l2h/img15.png "$ \operatorname{E}[\boldsymbol{X}]=\boldsymbol{\mu}$") ;
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![$ \operatorname{Var}[\boldsymbol{X}]=\boldsymbol{\Sigma}$](http://images.planetmath.org:8080/cache/objects/7204/l2h/img16.png "$ \operatorname{Var}[\boldsymbol{X}]=\boldsymbol{\Sigma}$") ;
- any linear combination of random vectors that are jointly normal is jointly normal. In fact, if
 an  dimensional random vector with a joint normal distribution,
 is an  constant real matrix, then
 (or
 depending on whether
 is a column or a row vector) is jointly normal;
- any marginal distribution of a joint normal distribution is jointly normal. In particular, if
 are jointly normal, then each  is normal;
- Let
 be a random vector whose distribution is jointly normal. Suppose coordinates of
 are partitioned into two groups, forming random vectors
 and
 , then the conditional distribution of
 given
 is jointly normal.
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"joint normal distribution" is owned by CWoo.
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(view preamble)
| Other names: |
multivariate Gaussian distribution |
| Also defines: |
jointly normal, multivariate normal distribution |
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Cross-references: conditional, groups, coordinates, distribution, normal, marginal distribution, row vector, column, linear combination, properties, simple, covariance matrix, mean vector, vector, matrix, real, positive definite, non-singular, random vector, Gaussian, probability density function, random variables, finite set
There are 5 references to this entry.
This is version 8 of joint normal distribution, born on 2005-07-01, modified 2006-12-06.
Object id is 7204, canonical name is JointNormalDistribution.
Accessed 13375 times total.
Classification:
| AMS MSC: | 60E05 (Probability theory and stochastic processes :: Distribution theory :: Distributions: general theory) | | | 62H05 (Statistics :: Multivariate analysis :: Characterization and structure theory) |
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Pending Errata and Addenda
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![$\displaystyle f_{\boldsymbol{X}}(\boldsymbol{x})=\sqrt{\frac{1 }{(2\pi)^p \det{... ...ratorname{T}}\boldsymbol{\Sigma}^{-1} (\boldsymbol{x} - \boldsymbol{\mu})\big],$](http://images.planetmath.org:8080/cache/objects/7204/l2h/img2.png "$\displaystyle f_{\boldsymbol{X}}(\boldsymbol{x})=\sqrt{\frac{1 }{(2\pi)^p \det{... ...ratorname{T}}\boldsymbol{\Sigma}^{-1} (\boldsymbol{x} - \boldsymbol{\mu})\big],$")
