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# 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. 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 $\boldsymbol{\Sigma}$.

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.

## Mathematics Subject Classification

62H05*no label found*60E05

*no label found*

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