Multivariate normal

Multivariate normal

Probability density function Many samples from a multivariate normal distribution with ${\displaystyle {\boldsymbol {\mu }}=\left[{\begin{smallmatrix}0\\0\end{smallmatrix}}\right]}$ and ${\displaystyle {\boldsymbol {\Sigma }}=\left[{\begin{smallmatrix}1&3/5\\3/5&2\end{smallmatrix}}\right]}$, shown along with the 3-sigma ellipse, the two marginal distributions, and the two 1-d histograms.
Notation	${\displaystyle {\mathcal {N}}({\boldsymbol {\mu }},\,{\boldsymbol {\Sigma }})}$
Parameters	μ ∈ Rk — location Σ ∈ Rk×k — covariance (positive-definite matrix)
Support	x ∈ μ + span(Σ) ⊆ Rk
PDF	${\displaystyle (2\pi )^{-{\frac {1}{2}}k}\left|{\boldsymbol {\Sigma }}\right|^{-{\frac {1}{2}}}\,e^{-{\frac {1}{2}}(\mathbf {x} -{\boldsymbol {\mu }})'{\boldsymbol {\Sigma }}^{-1}(\mathbf {x} -{\boldsymbol {\mu }})},}$ exists only when Σ is positive-definite
Mean	μ
Mode	μ
Variance	Σ
Entropy	${\displaystyle {\frac {1}{2}}\ln \left((2\pi e)^{k}\left|{\boldsymbol {\Sigma }}\right|\right)}$
MGF	${\displaystyle \exp \!{\Big (}{\boldsymbol {\mu }}'\mathbf {t} +{\tfrac {1}{2}}\mathbf {t} '{\boldsymbol {\Sigma }}\mathbf {t} {\Big )}}$
CF	${\displaystyle \exp \!{\Big (}i{\boldsymbol {\mu }}'\mathbf {t} -{\tfrac {1}{2}}\mathbf {t} '{\boldsymbol {\Sigma }}\mathbf {t} {\Big )}}$

In probability theory and statistics, the multivariate normal distribution or multivariate Gaussian distribution, is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One possible definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables each of which clusters around a mean value.

The multivariate normal distribution of a k-dimensional random vector x = [X₁, X₂, …, X_k]^T can be written in the following notation:

or to make it explicitly known that X is k-dimensional,

with k-dimensional mean vector

and ${\displaystyle k\times k}$ covariance matrix

A random vector x = (X₁, …, X_k)' is said to have the multivariate normal distribution if it satisfies the following equivalent conditions.