joint normal distribution

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

\lambda_{1}X_{1}+\lambda_{2}X_{2}+\cdots+\lambda_{n}X_{n}

are normal (http://planetmath.org/NormalRandomVariable). This implies, in particular, that the individual random variables $X_{i}$ 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 $\boldsymbol{\Sigma}$ ,

	$\displaystyle\mu_{i}=\mathbb{E}[X_{i}],$
	$\displaystyle\Sigma_{ij}=\operatorname{Cov}(X_{i},X_{j})=\mathbb{E}[X_{i}X_{j}% ]-\mathbb{E}[X_{i}]\mathbb{E}[X_{j}].$

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 $X, Y$ with $\operatorname{Var}(X)=2$ , $\operatorname{Var}(Y)=1$ and $\operatorname{Cov}(X,Y)=-1$ .

The joint normal distribution has the following properties:

1.

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

$f_{\boldsymbol{X}}(\boldsymbol{x})=\frac{1}{\sqrt{(2\pi)^{n}\det{\boldsymbol{(% \Sigma})}}}\exp\left(-\frac{1}{2}(\boldsymbol{x}-\boldsymbol{\mu})^{% \operatorname{T}}\boldsymbol{\Sigma}^{-1}(\boldsymbol{x}-\boldsymbol{\mu})% \right).$
2.

If $\boldsymbol{X}$ has the $\operatorname{N}(\boldsymbol{\mu},\boldsymbol{\Sigma})$ distribution and $\boldsymbol{\lambda}\in\mathbb{R}^{n}$ then

$\boldsymbol{\lambda}\cdot\boldsymbol{X}=\lambda_{1}X_{1}+\cdots+\lambda_{n}X_{% n}\sim\operatorname{N}(\boldsymbol{\lambda}\cdot\boldsymbol{\mu},\boldsymbol{% \lambda}^{\operatorname{T}}\boldsymbol{\Sigma}\boldsymbol{\lambda}).$
3.

Sets of linear combinations of joint normals are themselves joint normal. In particular, if $\boldsymbol{X}\sim\operatorname{N}(\boldsymbol{\mu},\boldsymbol{\Sigma})$ and $A$ is an $m\times n$ matrix, then $A\boldsymbol{X}$ has the joint normal distribution $\operatorname{N}(A\boldsymbol{\mu},A\boldsymbol{\Sigma}A^{\operatorname{T}})$ .
4.

The characteristic function is given by

$\varphi_{\boldsymbol{X}}(\boldsymbol{a})\equiv\mathbb{E}\left[\exp(i% \boldsymbol{a}\cdot\boldsymbol{X})\right]=\exp\left(i\boldsymbol{a}\cdot% \boldsymbol{\mu}-\frac{1}{2}\boldsymbol{a}^{\operatorname{T}}\boldsymbol{% \Sigma}\boldsymbol{a}\right),$

for $\boldsymbol{X}\sim\operatorname{N}(\boldsymbol{\mu},\boldsymbol{\Sigma})$ and any $\boldsymbol{a}\in\mathbb{C}^{n}$ .
5.

A pair $X, Y$ of jointly normal random variables are independent if and only if they have zero covariance.
6.

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.

Title	joint normal distribution
Canonical name	JointNormalDistribution
Date of creation	2013-03-22 15:22:34
Last modified on	2013-03-22 15:22:34
Owner	gel (22282)
Last modified by	gel (22282)
Numerical id	14
Author	gel (22282)
Entry type	Definition
Classification	msc 62H05
Classification	msc 60E05
Synonym	multivariate Gaussian distribution
Related topic	NormalRandomVariable
Defines	jointly normal
Defines	multivariate normal distribution