PlanetMath: joint normal distribution

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joint normal distribution

(Definition)

A finite set of random variables $X_1,\ldots,X_n$ are said to have a joint normal distribution or multivariate normal distribution if all real linear combinations

$\displaystyle \lambda_1X_1 + \lambda_2X_2 +\cdots+\lambda_nX_n$

are normal. This implies, in particular, that the individual random variables

are each normally distributed. However, the converse is not not true and sets of normally distributed random variables need not, in general, be jointly normal.

If $\boldsymbol{X}=(X_1,X_2,\ldots,X_n)$ is joint normal, then its probability distribution is uniquely determined by the means $\boldsymbol{\mu}\in\mathbb{R}^n$ and the $n\times n$ positive semidefinite covariance matrix $\boldsymbol{\Sigma}$ ,

	$\displaystyle \mu_i=\mathbb{E}[X_i],$
	$\displaystyle \Sigma_{ij}=\operatorname{Cov}(X_i,X_j)=\mathbb{E}[X_iX_j]-\mathbb{E}[X_i]\mathbb{E}[X_j].$

Then, the joint normal distribution is commonly denoted as $\operatorname{N}(\boldsymbol{\mu},\boldsymbol{\Sigma})$ . Conversely, this distribution exists for any such $\boldsymbol{\mu}$ and $\boldsymbol{\Sigma}$ .

**Figure 1:** Density of joint normal variables with $\operatorname{Var}(X)=2$ , $\operatorname{Var}(Y)=1$ and $\operatorname{Cov}(X,Y)=-1$ .
$\includegraphics[bb = 50 568 320 770,clip=,scale=1.2]{jointnormal}$

The joint normal distribution has the following properties:

If $\boldsymbol{X}$ has the $\operatorname{N}(\boldsymbol{\mu},\boldsymbol{\Sigma})$ distribution for nonsigular $\boldsymbol{\Sigma}$ then it has the multidimensional Gaussian probability density function

$\displaystyle f_{\boldsymbol{X}}(\boldsymbol{x})=\frac{1 }{\sqrt{(2\pi)^n \det{... ...torname{T}}\boldsymbol{\Sigma}^{-1} (\boldsymbol{x} - \boldsymbol{\mu})\right).$
If $\boldsymbol{X}$ has the $\operatorname{N}(\boldsymbol{\mu},\boldsymbol{\Sigma})$ distribution and $\boldsymbol{\lambda}\in\mathbb{R}^n$ then

$\displaystyle \boldsymbol{\lambda}\cdot\boldsymbol{X}=\lambda_1X_1+\cdots+\lamb... ...boldsymbol{\lambda}^{\operatorname{T}}\boldsymbol{\Sigma}\boldsymbol{\lambda}).$
Sets of linear combinations of joint normals are themselves joint normal. In particular, if $\boldsymbol{X}\sim\operatorname{N}(\boldsymbol{\mu},\boldsymbol{\Sigma})$ and is an $m\times n$ matrix, then $A\boldsymbol{X}$ has the joint normal distribution $\operatorname{N}(A\boldsymbol{\mu},A\boldsymbol{\Sigma}A^{\operatorname{T}})$ .
The characteristic function is given by

$\displaystyle \varphi_{\boldsymbol{X}}(\boldsymbol{a})\equiv\mathbb{E}\left[\ex... ...1}{2}\boldsymbol{a}^{\operatorname{T}}\boldsymbol{\Sigma}\boldsymbol{a}\right),$

for $\boldsymbol{X}\sim\operatorname{N}(\boldsymbol{\mu},\boldsymbol{\Sigma})$ and any $\boldsymbol{a}\in\mathbb{C}^n$ .
A pair of jointly normal random variables are independent if and only if they have zero covariance.
Let $\boldsymbol{X}$ be a random vector whose distribution is jointly normal. Suppose the coordinates of $\boldsymbol{X}$ are partitioned into two groups, forming random vectors $\boldsymbol{X_1}$ and $\boldsymbol{X_2}$ , then the conditional distribution of $\boldsymbol{X_1}$ given $\boldsymbol{X_2}=\boldsymbol{c}$ is jointly normal.

"joint normal distribution" is owned by gel. [ full author list (2) | owner history (1) ]

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See Also: normal random variable

Other names:

multivariate Gaussian distribution

Also defines:

jointly normal, multivariate normal distribution

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Cross-references: conditional, groups, coordinates, random vector, covariance, independent, characteristic function, matrix, normals, probability density function, Gaussian, properties, conversely, covariance matrix, positive semidefinite, distribution, converse, implies, linear combinations, real, random variables, finite set
There are 7 references to this entry.

This is version 11 of joint normal distribution, born on 2005-07-01, modified 2009-01-17.
Object id is 7204, canonical name is JointNormalDistribution.
Accessed 18829 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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