joint normal distribution
A finite set^{} of random variables^{} ${X}_{1},\mathrm{\dots},{X}_{n}$ are said to have a joint normal distribution or multivariate normal distribution if all real linear combinations^{}
$${\lambda}_{1}{X}_{1}+{\lambda}_{2}{X}_{2}+\mathrm{\cdots}+{\lambda}_{n}{X}_{n}$$ 
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 $\bm{X}=({X}_{1},{X}_{2},\mathrm{\dots},{X}_{n})$ is joint normal, then its probability distribution is uniquely determined by the means $\bm{\mu}\in {\mathbb{R}}^{n}$ and the $n\times n$ positive semidefinite covariance matrix^{} $\mathbf{\Sigma}$,
${\mu}_{i}=\mathbb{E}[{X}_{i}],$  
${\mathrm{\Sigma}}_{ij}=\mathrm{Cov}({X}_{i},{X}_{j})=\mathbb{E}[{X}_{i}{X}_{j}]\mathbb{E}[{X}_{i}]\mathbb{E}[{X}_{j}].$ 
Then, the joint normal distribution is commonly denoted as $\mathrm{N}(\bm{\mu},\mathbf{\Sigma})$. Conversely, this distribution^{} exists for any such $\bm{\mu}$ and $\mathbf{\Sigma}$.
The joint normal distribution has the following properties:

1.
If $\bm{X}$ has the $\mathrm{N}(\bm{\mu},\mathbf{\Sigma})$ distribution for nonsigular $\mathbf{\Sigma}$ then it has the multidimensional Gaussian probability density function
$${f}_{\bm{X}}(\bm{x})=\frac{1}{\sqrt{{(2\pi )}^{n}det\mathbf{(}\mathbf{\Sigma})}}\mathrm{exp}\left(\frac{1}{2}{(\bm{x}\bm{\mu})}^{\mathrm{T}}{\mathbf{\Sigma}}^{1}(\bm{x}\bm{\mu})\right).$$ 
2.
If $\bm{X}$ has the $\mathrm{N}(\bm{\mu},\mathbf{\Sigma})$ distribution and $\bm{\lambda}\in {\mathbb{R}}^{n}$ then
$$\bm{\lambda}\cdot \bm{X}={\lambda}_{1}{X}_{1}+\mathrm{\cdots}+{\lambda}_{n}{X}_{n}\sim \mathrm{N}(\bm{\lambda}\cdot \bm{\mu},{\bm{\lambda}}^{\mathrm{T}}\mathbf{\Sigma}\bm{\lambda}).$$ 
3.
Sets of linear combinations of joint normals are themselves joint normal. In particular, if $\bm{X}\sim \mathrm{N}(\bm{\mu},\mathbf{\Sigma})$ and $A$ is an $m\times n$ matrix, then $A\bm{X}$ has the joint normal distribution $\mathrm{N}(A\bm{\mu},A\mathbf{\Sigma}{A}^{\mathrm{T}})$.

4.
The characteristic function^{} is given by
$${\phi}_{\bm{X}}(\bm{a})\equiv \mathbb{E}\left[\mathrm{exp}(i\bm{a}\cdot \bm{X})\right]=\mathrm{exp}\left(i\bm{a}\cdot \bm{\mu}\frac{1}{2}{\bm{a}}^{\mathrm{T}}\mathbf{\Sigma}\bm{a}\right),$$ for $\bm{X}\sim \mathrm{N}(\bm{\mu},\mathbf{\Sigma})$ and any $\bm{a}\in {\u2102}^{n}$.

5.
A pair $X,Y$ of jointly normal random variables are independent if and only if they have zero covariance^{}.

6.
Let $\bm{X}$ be a random vector whose distribution is jointly normal. Suppose the coordinates^{} of $\bm{X}$ are partitioned into two groups, forming random vectors ${\bm{X}}_{\mathrm{\U0001d7cf}}$ and ${\bm{X}}_{\mathrm{\U0001d7d0}}$, then the conditional distribution of ${\bm{X}}_{\mathrm{\U0001d7cf}}$ given ${\bm{X}}_{\mathrm{\U0001d7d0}}=\bm{c}$ is jointly normal.
Title  joint normal distribution 

Canonical name  JointNormalDistribution 
Date of creation  20130322 15:22:34 
Last modified on  20130322 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 