Multivariate normal distribution‎

The multivariate normal distribution of a k-dimensional random vector $\backslash mathbf\{X\}\; =\; (X\_1,\backslash ldots,X\_k)^\{\backslash mathrm\; T\}$ can be written in the following notation:

: $$

\mathbf{X}\ \sim\ \mathcal{N}(\boldsymbol\mu,\, \boldsymbol\Sigma),

or to make it explicitly known that $\backslash mathbf\{X\}$ is k-dimensional,

: $$

\mathbf{X}\ \sim\ \mathcal{N}_k(\boldsymbol\mu,\, \boldsymbol\Sigma),

with k-dimensional mean vector

:$\backslash boldsymbol\backslash mu\; =\; \backslash operatorname\{E\}[\backslash mathbf\{X\}]\; =\; (\; \backslash operatorname\{E\}[X\_1],\; \backslash operatorname\{E\}[X\_2],\; \backslash ldots,\; \backslash operatorname\{E\}[X\_k]\; )\; ^\; \backslash mathrm\{T\},$

and $k\; \backslash times\; k$ covariance matrix

:$\backslash Sigma\_\{i,j\}\; =\; \backslash operatorname\{E\}\; [(X\_i\; -\; \backslash mu\_i)(\; X\_j\; -\; \backslash mu\_j)]\; =\; \backslash operatorname\{Cov\}[X\_i,\; X\_j]$

such that $1\; \backslash le\; i\; \backslash le\; k$ and $1\; \backslash le\; j\; \backslash le\; k$. The inverse of the covariance matrix is called the precision matrix, denoted by $\backslash boldsymbol\{Q\}=\backslash boldsymbol\backslash Sigma^\{-1\}$.

A real random vector $\backslash mathbf\{X\}\; =\; (X\_1,\backslash ldots,X\_k)^\{\backslash mathrm\; T\}$ is called a standard normal random vector if all of its components $X\_i$ are independent and each is a zero-mean unit-variance normally distributed random variable, i.e. if $X\_i\; \backslash sim\backslash \; \backslash mathcal\{N\}(0,1)$ for all $i=1\backslash ldots\; k$.{{cite book |first=Amos |last=Lapidoth |year=2009 |title=A Foundation in Digital Communication |publisher=Cambridge University Press |isbn=978-0-521-19395-5}}{{rp|p. 454}}

A real random vector $\backslash mathbf\{X\}\; =\; (X\_1,\backslash ldots,X\_k)^\{\backslash mathrm\; T\}$ is called a centered normal random vector if there exists a $k\; \backslash times\; \backslash ell$ matrix $\backslash boldsymbol\{A\}$ such that $\backslash boldsymbol\{A\}\; \backslash mathbf\{Z\}$ has the same distribution as $\backslash mathbf\{X\}$ where $\backslash mathbf\{Z\}$ is a standard normal random vector with $\backslash ell$ components.{{rp|p. 454}}

A real random vector $\backslash mathbf\{X\}\; =\; (X\_1,\backslash ldots,X\_k)^\{\backslash mathrm\; T\}$ is called a normal random vector if there exists a random $\backslash ell$-vector $\backslash mathbf\{Z\}$, which is a standard normal random vector, a $k$-vector $\backslash boldsymbol\backslash mu$, and a $k\; \backslash times\; \backslash ell$ matrix $\backslash boldsymbol\{A\}$, such that $\backslash mathbf\{X\}=\backslash boldsymbol\{A\}\; \backslash mathbf\{Z\}\; +\; \backslash boldsymbol\backslash mu$.{{cite book |first=Allan |last=Gut |year=2009 |title=An Intermediate Course in Probability |publisher=Springer |isbn=978-1-441-90161-3}}{{rp|p. 454}}{{rp|p. 455}}

Formally:

{{Equation box 1

|indent =

|title=

|equation =

\mathbf{X}\ \sim\ \mathcal{N}_k(\boldsymbol\mu, \boldsymbol\Sigma) \iff \text{there exist } \boldsymbol\mu\in\mathbb{R}^k,\boldsymbol{A}\in\mathbb{R}^{k\times \ell} \text{ such that } \mathbf{X}=\boldsymbol{A} \mathbf{Z} + \boldsymbol\mu \text{ and } \forall n=1,\ldots,\ell : Z_n \sim\ \mathcal{N}(0, 1), \text{i.i.d.}

|cellpadding= 6

|border

|border colour = #0073CF

|background colour=#F5FFFA}}

Here the covariance matrix is $\backslash boldsymbol\backslash Sigma\; =\; \backslash boldsymbol\{A\}\; \backslash boldsymbol\{A\}^\{\backslash mathrm\; T\}$.

In the degenerate case where the covariance matrix is singular, the corresponding distribution has no density; see the section below for details. This case arises frequently in statistics; for example, in the distribution of the vector of residuals in the ordinary least squares regression. The $X\_i$ are in general not independent; they can be seen as the result of applying the matrix $\backslash boldsymbol\{A\}$ to a collection of independent Gaussian variables $\backslash mathbf\{Z\}$.

The following definitions are equivalent to the definition given above. A random vector $\backslash mathbf\{X\}\; =\; (X\_1,\; \backslash ldots,\; X\_k)^\backslash mathrm\{T\}$ has a multivariate normal distribution if it satisfies one of the following equivalent conditions.

• Every linear combination $Y=a\_1\; X\_1\; +\; \backslash cdots\; +\; a\_k\; X\_k$ of its components is normally distributed. That is, for any constant vector $\backslash mathbf\{a\}\; \backslash in\; \backslash mathbb\{R\}^k$, the random variable $Y=\backslash mathbf\{a\}^\{\backslash mathrm\; T\}\backslash mathbf\{X\}$ has a univariate normal distribution, where a univariate normal distribution with zero variance is a point mass on its mean.
• There is a k-vector $\backslash mathbf\{\backslash mu\}$ and a symmetric, positive semidefinite $k\; \backslash times\; k$ matrix $\backslash boldsymbol\backslash Sigma$, such that the characteristic function of $\backslash mathbf\{X\}$ is $$$$

\varphi_\mathbf{X}(\mathbf{u}) = \exp\Big( i\mathbf{u}^\mathrm{T}\boldsymbol\mu - \tfrac{1}{2} \mathbf{u}^\mathrm{T}\boldsymbol\Sigma \mathbf{u} \Big).

The spherical normal distribution can be characterised as the unique distribution where components are independent in any orthogonal coordinate system.{{cite journal |last1=Kac |first1=M. |title=On a characterization of the normal distribution |journal=American Journal of Mathematics |date=1939 |volume=61 |issue=3 |pages=726–728 |jstor=2371328 |doi=10.2307/2371328 }}{{cite journal |last1=Sinz |first1=Fabian |last2=Gerwinn |first2=Sebastian |last3=Bethge |first3=Matthias |title=Characterization of the p-generalized normal distribution |journal=Journal of Multivariate Analysis |date=2009 |volume=100 |issue=5 |pages=817–820 |doi=10.1016/j.jmva.2008.07.006 |doi-access=free }}

File:Multivariate Gaussian.png]]

The multivariate normal distribution is said to be "non-degenerate" when the symmetric covariance matrix $\backslash boldsymbol\backslash Sigma$ is positive definite. In this case the distribution has densitySimon J.D. Prince(June 2012). [http://www.computervisionmodels.com/ Computer Vision: Models, Learning, and Inference] {{Webarchive|url=https://web.archive.org/web/20201028120753/http://computervisionmodels.com/ |date=2020-10-28 }}. Cambridge University Press. 3.7:"Multivariate normal distribution".

{{Equation box 1

|indent =

|title=

|equation = $$

f_{\mathbf X}(x_1,\ldots,x_k) = \frac{\exp\left(-\frac 1 2 \left({\mathbf x} - {\boldsymbol\mu}\right)^\mathrm{T}{\boldsymbol\Sigma}^{-1}\left({\mathbf x}-{\boldsymbol\mu}\right)\right)}{\sqrt{(2\pi)^k |\boldsymbol\Sigma|}}

|cellpadding= 6

|border

|border colour = #0073CF

|background colour=#F5FFFA}}

where $\{\backslash mathbf\; x\}$ is a real k-dimensional column vector and $|\backslash boldsymbol\backslash Sigma|\backslash equiv\; \backslash det\backslash boldsymbol\backslash Sigma$ is the determinant of $\backslash boldsymbol\backslash Sigma$, also known as the generalized variance. The equation above reduces to that of the univariate normal distribution if $\backslash boldsymbol\backslash Sigma$ is a $1\; \backslash times\; 1$ matrix (i.e., a single real number).

The circularly symmetric version of the complex normal distribution has a slightly different form.

Each iso-density locus — the locus of points in k-dimensional space each of which gives the same particular value of the density — is an ellipse or its higher-dimensional generalization; hence the multivariate normal is a special case of the elliptical distributions.

The quantity $\backslash sqrt\{(\{\backslash mathbf\; x\}-\{\backslash boldsymbol\backslash mu\})^\backslash mathrm\{T\}\{\backslash boldsymbol\backslash Sigma\}^\{-1\}(\{\backslash mathbf\; x\}-\{\backslash boldsymbol\backslash mu\})\}$ is known as the Mahalanobis distance, which represents the distance of the test point $\{\backslash mathbf\; x\}$ from the mean $\{\backslash boldsymbol\backslash mu\}$.

The squared Mahalanobis distance

$(\{\backslash mathbf\; x\}-\{\backslash boldsymbol\backslash mu\})^\backslash mathrm\{T\}\{\backslash boldsymbol\backslash Sigma\}^\{-1\}(\{\backslash mathbf\; x\}-\{\backslash boldsymbol\backslash mu\})$ is decomposed into a sum of k terms, each term being a product of three meaningful components.{{Cite journal |last=Kim |first=M. G. |year=2000 |title=Multivariate outliers and decompositions of Mahalanobis distance |journal=Communications in Statistics – Theory and Methods |volume=29 |issue=7 |pages=1511–1526 |doi=10.1080/03610920008832559}}

Note that in the case when $k\; =\; 1$, the distribution reduces to a univariate normal distribution and the Mahalanobis distance reduces to the absolute value of the standard score. See also Interval below.

In the 2-dimensional nonsingular case ($k\; =\; \backslash operatorname\{rank\}\backslash left(\backslash Sigma\backslash right)\; =\; 2$), the probability density function of a vector $\backslash text\{[XY]\}\backslash prime$ is:

f(x,y) =

\frac{1}{2 \pi \sigma_X \sigma_Y \sqrt{1-\rho^2}}

\exp

\left( -\frac{1}{2\left[1 - \rho^2\right]}\left[

\left(\frac{x-\mu_X}{\sigma_X}\right)^2 -

2\rho\left(\frac{x - \mu_X}{\sigma_X}\right)\left(\frac{y - \mu_Y}{\sigma_Y}\right) +

\left(\frac{y - \mu_Y}{\sigma_Y}\right)^2

\right]

\right)

where $\backslash rho$ is the correlation between $X$ and $Y$ and

where $\backslash sigma\_X>0$ and $\backslash sigma\_Y>0$. In this case,

: $$

\boldsymbol\mu = \begin{pmatrix} \mu_X \\ \mu_Y \end{pmatrix}, \quad

\boldsymbol\Sigma = \begin{pmatrix} \sigma_X^2 & \rho \sigma_X \sigma_Y \\

\rho \sigma_X \sigma_Y & \sigma_Y^2 \end{pmatrix}.

In the bivariate case, the first equivalent condition for multivariate reconstruction of normality can be made less restrictive as it is sufficient to verify that a countably infinite set of distinct linear combinations of $X$ and $Y$ are normal in order to conclude that the vector of $\backslash text\{[XY]\}\backslash prime$ is bivariate normal.

The bivariate iso-density loci plotted in the $x,y$-plane are ellipses, whose principal axes are defined by the eigenvectors of the covariance matrix $\backslash boldsymbol\backslash Sigma$ (the major and minor semidiameters of the ellipse equal the square-root of the ordered eigenvalues).

File:GaussianScatterPCA.png

As the absolute value of the correlation parameter $\backslash rho$ increases, these loci are squeezed toward the following line :

: $$

y(x) = \sgn (\rho)\frac{\sigma_Y}{\sigma_X} (x - \mu _X) + \mu_Y.

This is because this expression, with $\backslash sgn(\backslash rho)$ (where sgn is the sign function) replaced by $\backslash rho$, is the best linear unbiased prediction of $Y$ given a value of $X$.

If the covariance matrix $\backslash boldsymbol\backslash Sigma$ is not full rank, then the multivariate normal distribution is degenerate and does not have a density. More precisely, it does not have a density with respect to k-dimensional Lebesgue measure (which is the usual measure assumed in calculus-level probability courses). Only random vectors whose distributions are absolutely continuous with respect to a measure are said to have densities (with respect to that measure). To talk about densities but avoid dealing with measure-theoretic complications it can be simpler to restrict attention to a subset of $\backslash operatorname\{rank\}(\backslash boldsymbol\backslash Sigma)$ of the coordinates of $\backslash mathbf\{x\}$ such that the covariance matrix for this subset is positive definite; then the other coordinates may be thought of as an affine function of these selected coordinates.{{Cite web |title=linear algebra - Mapping between affine coordinate function |url=https://math.stackexchange.com/q/2727720 |access-date=2022-06-24 |website=Mathematics Stack Exchange |language=en}}

To talk about densities meaningfully in singular cases, then, we must select a different base measure. Using the disintegration theorem we can define a restriction of Lebesgue measure to the $\backslash operatorname\{rank\}(\backslash boldsymbol\backslash Sigma)$-dimensional affine subspace of $\backslash mathbb\{R\}^k$ where the Gaussian distribution is supported, i.e. $\backslash left\backslash \{\backslash boldsymbol\backslash mu\; +\; \backslash boldsymbol\{\backslash Sigma^\{1/2\}\}\backslash mathbf\{v\}\; :\; \backslash mathbf\{v\}\; \backslash in\; \backslash mathbb\{R\}^k\backslash right\backslash \}$. With respect to this measure the distribution has the density of the following motif:

:$f(\backslash mathbf\{x\})\; =\; \backslash frac\{\backslash exp\backslash left(\; -\backslash frac\{1\}\{2\}\backslash left(\backslash mathbf\{x\}\; -\; \backslash boldsymbol\backslash mu\backslash right)^\backslash mathrm\{T\}\; \backslash boldsymbol\backslash Sigma^+\backslash left(\backslash mathbf\{x\}\; -\; \backslash boldsymbol\backslash mu\backslash right)\backslash right)\; \}\backslash sqrt\{\backslash det\backslash nolimits^*(2\backslash pi\backslash boldsymbol\backslash Sigma)\}$

where $\backslash boldsymbol\backslash Sigma^+$ is the generalized inverse and $\backslash det\backslash nolimits^*$ is the pseudo-determinant.

The notion of cumulative distribution function (cdf) in dimension 1 can be extended in two ways to the multidimensional case, based on rectangular and ellipsoidal regions.

The first way is to define the cdf $F(\backslash mathbf\{x\})$ of a random vector $\backslash mathbf\{X\}$ as the probability that all components of $\backslash mathbf\{X\}$ are less than or equal to the corresponding values in the vector $\backslash mathbf\{x\}$:{{cite journal|last1=Botev|first1=Z. I.|title=The normal law under linear restrictions: simulation and estimation via minimax tilting|journal=Journal of the Royal Statistical Society, Series B|volume=79|pages=125–148|date=2016|doi=10.1111/rssb.12162|arxiv=1603.04166|bibcode=2016arXiv160304166B|s2cid=88515228}}

:$F(\backslash mathbf\{x\})\; =\; \backslash mathbb\{P\}(\backslash mathbf\{X\}\backslash leq\; \backslash mathbf\{x\}),\; \backslash quad\; \backslash text\{where\; \}\; \backslash mathbf\{X\}\; \backslash sim\; \backslash mathcal\{N\}(\backslash boldsymbol\backslash mu,\backslash ,\; \backslash boldsymbol\backslash Sigma).$

Though there is no closed form for $F(\backslash mathbf\{x\})$, there are a number of algorithms that estimate it numerically.{{cite book|last=Genz|first=Alan|title=Computation of Multivariate Normal and t Probabilities|date=2009|publisher=Springer|isbn=978-3-642-01689-9|url=https://www.springer.com/statistics/computational+statistics/book/978-3-642-01688-2}}

Another way is to define the cdf $F(r)$ as the probability that a sample lies inside the ellipsoid determined by its Mahalanobis distance $r$ from the Gaussian, a direct generalization of the standard deviation.[https://upload.wikimedia.org/wikipedia/commons/a/a2/Cumulative_function_n_dimensional_Gaussians_12.2013.pdf Bensimhoun Michael, N-Dimensional Cumulative Function, And Other Useful Facts About Gaussians and Normal Densities (2006)]

In order to compute the values of this function, closed analytic formula exist, as follows.

{{further|Confidence region|Hotelling t-squared statistic}}

The interval for the multivariate normal distribution yields a region consisting of those vectors x satisfying

:$(\{\backslash mathbf\; x\}-\{\backslash boldsymbol\backslash mu\})^\backslash mathrm\{T\}\{\backslash boldsymbol\backslash Sigma\}^\{-1\}(\{\backslash mathbf\; x\}-\{\backslash boldsymbol\backslash mu\})\; \backslash leq\; \backslash chi^2\_k(p).$

Here $\{\backslash mathbf\; x\}$ is a $k$-dimensional vector, $\{\backslash boldsymbol\backslash mu\}$ is the known $k$-dimensional mean vector, $\backslash boldsymbol\backslash Sigma$ is the known covariance matrix and $\backslash chi^2\_k(p)$ is the quantile function for probability $p$ of the chi-squared distribution with $k$ degrees of freedom.

When $k\; =\; 2,$ the expression defines the interior of an ellipse and the chi-squared distribution simplifies to an exponential distribution with mean equal to two (rate equal to half).

The complementary cumulative distribution function (ccdf) or the tail distribution

is defined as $\backslash overline\; F(\backslash mathbf\{x\})\; =\; 1\; -\; \backslash mathbb\; P\backslash left(\backslash mathbf\; X\backslash leq\; \backslash mathbf\; x\backslash right)$.

When $\backslash mathbf\{X\}\; \backslash sim\; \backslash mathcal\{N\}(\backslash boldsymbol\backslash mu,\backslash ,\; \backslash boldsymbol\backslash Sigma)$, then

the ccdf can be written as a probability the maximum of dependent Gaussian variables:{{cite conference |title=Tail distribution of the maximum of correlated Gaussian random variables |last1=Botev |first1=Z. I. |last2=Mandjes |first2=M. |last3=Ridder |first3=A. |publisher=IEEE|isbn=978-1-4673-9743-8 |book-title= 2015 Winter Simulation Conference (WSC) |pages=633–642 |location=Huntington Beach, Calif., USA |date=6–9 December 2015 |doi= 10.1109/WSC.2015.7408202 |hdl=10419/130486 |hdl-access=free }}

:$\backslash overline\; F(\backslash mathbf\{x\})\; =\; \backslash mathbb\; P\backslash left(\backslash bigcup\_i\backslash \{X\_i\backslash geq\; x\_i\backslash \}\backslash right)\; =\; \backslash mathbb\; P\backslash left(\backslash max\_i\; Y\_i\backslash geq\; 0\backslash right),$

\quad \text{where } \mathbf{Y} \sim \mathcal{N}\left(\boldsymbol\mu - \mathbf{x},\, \boldsymbol\Sigma\right).

While no simple closed formula exists for computing the ccdf, the maximum of dependent Gaussian variables can

be estimated accurately via the Monte Carlo method.{{cite conference |title=Efficient simulation for tail probabilities of Gaussian random fields |last1=Adler |first1=R. J.

|last2=Blanchet |first2=J. |last3=Liu |first3=J. |publisher=IEEE|isbn=978-1-4244-2707-9

|book-title= 2008 Winter Simulation Conference (WSC) |pages=328–336 |location=Miami, Fla., USA |date=7–10 Dec 2008

|doi= 10.1109/WSC.2008.4736085 }}

File:Multivariate normal probability in different domains.png

The probability content of the multivariate normal in a quadratic domain defined by $q(\backslash boldsymbol\{x\})\; =\; \backslash boldsymbol\{x\}\text{'}\; \backslash mathbf\{Q\_2\}\; \backslash boldsymbol\{x\}\; +\; \backslash boldsymbol\{q\_1\}\text{'}\; \backslash boldsymbol\{x\}\; +\; q\_0>0$ (where $\backslash mathbf\{Q\_2\}$ is a matrix, $\backslash boldsymbol\{q\_1\}$ is a vector, and $q\_0$ is a scalar), which is relevant for Bayesian classification/decision theory using Gaussian discriminant analysis, is given by the generalized chi-squared distribution.{{cite arXiv |eprint=2012.14331 |last1=Das |first1=Abhranil |author2=Wilson S Geisler |title=Methods to integrate multinormals and compute classification measures |date=2020 |class=stat.ML }}

The probability content within any general domain defined by $f(\backslash boldsymbol\{x\})>0$ (where $f(\backslash boldsymbol\{x\})$ is a general function) can be computed using the numerical method of ray-tracing ([https://www.mathworks.com/matlabcentral/fileexchange/84973-integrate-and-classify-normal-distributions Matlab code]).

{{Main|Isserlis' theorem}}

The kth-order moments of x are given by

:$\backslash mu\_\{1,\backslash ldots,N\}(\backslash mathbf\{x\})$

\mathrel\stackrel{\mathrm{def}}{=} \mu _{r_1,\ldots,r_N}(\mathbf{x})

\mathrel\stackrel{\mathrm{def}}{=} \operatorname E\left[ \prod_{j=1}^N X_j^{r_j} \right]

where {{math|r₁ + r₂ + ⋯ + r_N {{=}} k.}}

The kth-order central moments are as follows

{{ordered list|list_style_type=lower-alpha

|If k is odd, {{math|μ_{1, ..., N}(x − μ) {{=}} 0}}.

|If k is even with {{math|k {{=}} 2λ}}, then{{Ambiguous|date=November 2022|reason=Unclear notation}} $$$$

\mu_{1,\dots,2\lambda}(\mathbf{x}-\boldsymbol\mu ) = \sum \left( \sigma_{ij} \sigma_{k\ell} \cdots \sigma_{XZ}\right)

}}

where the sum is taken over all allocations of the set $\backslash left\backslash \{\; 1,\backslash ldots,2\backslash lambda\; \backslash right\backslash \}$ into λ (unordered) pairs. That is, for a kth {{math| ({{=}} 2λ {{=}} 6)}} central moment, one sums the products of {{nowrap|λ {{=}} 3}} covariances (the expected value μ is taken to be 0 in the interests of parsimony):

: $\backslash begin\{align\}$

& \operatorname E[X_1 X_2 X_3 X_4 X_5 X_6] \\[8pt]

= {} & \operatorname E[X_1 X_2]\operatorname E[X_3 X_4]\operatorname E[X_5 X_6] + \operatorname E[X_1 X_2]\operatorname E[X_3 X_5]\operatorname E[X_4 X_6] + \operatorname E[X_1 X_2]\operatorname E[X_3 X_6] \operatorname E[X_4 X_5] \\[4pt]

& {} + \operatorname E[X_1 X_3]\operatorname E[X_2 X_4]\operatorname E[X_5 X_6] + \operatorname E[X_1 X_3]\operatorname E[X_2 X_5]\operatorname E[X_4 X_6] + \operatorname E[X_1 X_3]\operatorname E[X_2 X_6] \operatorname E[X_4 X_5] \\[4pt]

& {} + \operatorname E[X_1 X_4]\operatorname E[X_2 X_3]\operatorname E[X_5 X_6] + \operatorname E[X_1 X_4]\operatorname E[X_2 X_5]\operatorname E[X_3 X_6]+ \operatorname E[X_1 X_4]\operatorname E[X_2 X_6] \operatorname E[X_3 X_5] \\[4pt]

& {} + \operatorname E[X_1 X_5]\operatorname E[X_2 X_3]\operatorname E[X_4 X_6] + \operatorname E[X_1 X_5]\operatorname E[X_2 X_4]\operatorname E[X_3 X_6] + \operatorname E[X_1 X_5]\operatorname E[X_2 X_6] \operatorname E[X_3 X_4] \\[4pt]

& {} + \operatorname E[X_1 X_6]\operatorname E[X_2 X_3]\operatorname E[X_4 X_5] + \operatorname E[X_1 X_6]\operatorname E[X_2 X_4]\operatorname E[X_3 X_5] + \operatorname E[X_1 X_6] \operatorname E[X_2 X_5]\operatorname E[X_3 X_4].

\end{align}

This yields $\backslash tfrac\{(2\backslash lambda\; -1)!\}\{2^\{\backslash lambda\; -1\}(\backslash lambda\; -1)!\}$ terms in the sum (15 in the above case), each being the product of λ (in this case 3) covariances. For fourth order moments (four variables) there are three terms. For sixth-order moments there are {{math|3 × 5 {{=}} 15}} terms, and for eighth-order moments there are {{math|3 × 5 × 7 {{=}} 105}} terms.

The covariances are then determined by replacing the terms of the list $[\; 1,\; \backslash ldots,\; 2\backslash lambda]$ by the corresponding terms of the list consisting of r₁ ones, then r₂ twos, etc.. To illustrate this, examine the following 4th-order central moment case:

: $$

\begin{align}

\operatorname E \left [ X_i^4 \right ] & = 3\sigma_{ii}^2 \\[4pt]

\operatorname E \left[ X_i^3 X_j \right ] & = 3\sigma_{ii} \sigma_{ij} \\[4pt]

\operatorname E \left[ X_i^2 X_j^2 \right ] & = \sigma_{ii}\sigma_{jj}+2 \sigma _{ij}^2 \\[4pt]

\operatorname E \left[ X_i^2 X_j X_k \right ] & = \sigma_{ii}\sigma _{jk}+2\sigma _{ij}\sigma_{ik} \\[4pt]

\operatorname E \left [ X_i X_j X_k X_n \right ] & = \sigma_{ij}\sigma _{kn} + \sigma _{ik} \sigma_{jn} + \sigma_{in} \sigma _{jk}.

\end{align}

where $\backslash sigma\_\{ij\}$ is the covariance of X_i and X_j. With the above method one first finds the general case for a kth moment with k different X variables, $E\backslash left[\; X\_i\; X\_j\; X\_k\; X\_n\backslash right]$, and then one simplifies this accordingly. For example, for $\backslash operatorname\; E[\; X\_i^2\; X\_k\; X\_n\; ]$, one lets {{math|X_i {{=}} X_j}} and one uses the fact that $\backslash sigma\_\{ii\}\; =\; \backslash sigma\_i^2$.

[[File:Probabilities of functions of normal vectors.png|thumb|right|a: Probability density of a function $\backslash cos\; x^2$ of a single normal variable $x$ with $\backslash mu=-2$ and $\backslash sigma=3$. b: Probability density of a function $x^y$ of a normal vector $(x,\; y)$, with mean $\backslash boldsymbol\{\backslash mu\}=(1,\; 2)$, and covariance

$\backslash mathbf\{\backslash Sigma\}\; =\; \backslash begin\{bmatrix\}$

.01 & .016 \\

.016 & .04

\end{bmatrix}. c: Heat map of the joint probability density of two functions of a normal vector $(x,\; y)$, with mean $\backslash boldsymbol\{\backslash mu\}=(-2,\; 5)$, and covariance

$\backslash mathbf\{\backslash Sigma\}\; =\; \backslash begin\{bmatrix\}$

10 & -7 \\

-7 & 10

\end{bmatrix}. d: Probability density of a function $\backslash sum\_\{i=1\}^4\; \backslash vert\; x\_i\; \backslash vert$ of 4 iid standard normal variables. These are computed by the numerical method of ray-tracing.]]

A quadratic form of a normal vector $\backslash boldsymbol\{x\}$, $q(\backslash boldsymbol\{x\})\; =\; \backslash boldsymbol\{x\}\text{'}\; \backslash mathbf\{Q\_2\}\; \backslash boldsymbol\{x\}\; +\; \backslash boldsymbol\{q\_1\}\text{'}\; \backslash boldsymbol\{x\}\; +\; q\_0$ (where $\backslash mathbf\{Q\_2\}$ is a matrix, $\backslash boldsymbol\{q\_1\}$ is a vector, and $q\_0$ is a scalar), is a generalized chi-squared variable. The direction of a normal vector follows a projected normal distribution.

If $f(\backslash boldsymbol\{x\})$ is a general scalar-valued function of a normal vector, its probability density function, cumulative distribution function, and inverse cumulative distribution function can be computed with the numerical method of ray-tracing ([https://www.mathworks.com/matlabcentral/fileexchange/84973-integrate-and-classify-normal-distributions Matlab code]).

If the mean and covariance matrix are known, the log likelihood of an observed vector $\backslash boldsymbol\{x\}$ is simply the log of the probability density function:

:$\backslash ln\; L\; (\backslash boldsymbol\{x\})=\; -\backslash frac\{1\}\{2\}\; \backslash left[\; \backslash ln\; (|\backslash boldsymbol\backslash Sigma|\backslash ,)\; +\; (\backslash boldsymbol\{x\}-\backslash boldsymbol\backslash mu)\text{'}\backslash boldsymbol\backslash Sigma^\{-1\}(\backslash boldsymbol\{x\}-\backslash boldsymbol\backslash mu)\; +\; k\backslash ln(2\backslash pi)\; \backslash right]$,

The circularly symmetric version of the noncentral complex case, where $\backslash boldsymbol\{z\}$ is a vector of complex numbers, would be

:$\backslash ln\; L\; (\backslash boldsymbol\{z\})\; =\; -\backslash ln\; (|\backslash boldsymbol\backslash Sigma|\backslash ,)\; -(\backslash boldsymbol\{z\}-\backslash boldsymbol\backslash mu)^\backslash dagger\backslash boldsymbol\backslash Sigma^\{-1\}(\backslash boldsymbol\{z\}-\backslash boldsymbol\backslash mu)\; -k\backslash ln(\backslash pi)$

i.e. with the conjugate transpose (indicated by $\backslash dagger$) replacing the normal transpose (indicated by $\text{'}$). This is slightly different than in the real case, because the circularly symmetric version of the complex normal distribution has a slightly different form for the normalization constant.

A similar notation is used for multiple linear regression.Tong, T. (2010) [http://amath.colorado.edu/courses/7400/2010Spr/lecture9.pdf Multiple Linear Regression : MLE and Its Distributional Results] {{webarchive|url=https://www.webcitation.org/6HPbX5thy?url=http://amath.colorado.edu/courses/7400/2010Spr/lecture9.pdf|date=2013-06-16}}, Lecture Notes

Since the log likelihood of a normal vector is a quadratic form of the normal vector, it is distributed as a generalized chi-squared variable.

The differential entropy of the multivariate normal distribution is{{cite journal

| last1 = Gokhale | first1 = DV

| last2 = Ahmed | first2 = NA

| last3 = Res |first3=BC

| last4 = Piscataway |first4=NJ

| date = May 1989

| title = Entropy Expressions and Their Estimators for Multivariate Distributions

| journal = IEEE Transactions on Information Theory

| volume = 35 | issue = 3 | pages = 688–692

| doi =10.1109/18.30996

}}

\begin{align}

h\left(f\right) & = -\int_{-\infty}^\infty \int_{-\infty}^\infty \cdots\int_{-\infty}^\infty f(\mathbf{x}) \ln f(\mathbf{x})\,d\mathbf{x} \\[1ex]

& = \frac12 \ln \left|2\pi e\boldsymbol\Sigma \right| = \frac{k}{2} \left(1 + \ln 2\pi\right) + \frac{1}{2} \ln \left|\boldsymbol\Sigma \right|,

\end{align}

where the bars denote the matrix determinant, {{math|k}} is the dimensionality of the vector space, and the result has units of nats.

The Kullback–Leibler divergence from $\backslash mathcal\{N\}\_1(\backslash boldsymbol\backslash mu\_1,\; \backslash boldsymbol\backslash Sigma\_1)$ to $\backslash mathcal\{N\}\_0(\backslash boldsymbol\backslash mu\_0,\; \backslash boldsymbol\backslash Sigma\_0)$, for non-singular matrices Σ₁ and Σ₀, is:{{cite thesis |first=J. |last=Duchi |title=Derivations for Linear Algebra and Optimization |url=https://stanford.edu/~jduchi/projects/general_notes.pdf#page=13 |page=13 }}

:$$

D_\text{KL}(\mathcal{N}_0 \parallel \mathcal{N}_1) = { 1 \over 2 } \left\{ \operatorname{tr} \left( \boldsymbol\Sigma_1^{-1} \boldsymbol\Sigma_0 \right) + \left( \boldsymbol\mu_1 - \boldsymbol\mu_0\right)^{\rm T} \boldsymbol\Sigma_1^{-1} ( \boldsymbol\mu_1 - \boldsymbol\mu_0 ) - k +\ln { | \boldsymbol \Sigma_1 | \over | \boldsymbol\Sigma_0 | } \right\},

where $|\backslash cdot|$ denotes the matrix determinant, $tr(\backslash cdot)$ is the trace, $ln(\backslash cdot)$ is the natural logarithm and $k$ is the dimension of the vector space.

The logarithm must be taken to base e since the two terms following the logarithm are themselves base-e logarithms of expressions that are either factors of the density function or otherwise arise naturally. The equation therefore gives a result measured in nats. Dividing the entire expression above by log_e 2 yields the divergence in bits.

When $\backslash boldsymbol\backslash mu\_1\; =\; \backslash boldsymbol\backslash mu\_0$,

:$$

D_\text{KL}(\mathcal{N}_0 \parallel \mathcal{N}_1) = { 1 \over 2 } \left\{ \operatorname{tr} \left( \boldsymbol\Sigma_1^{-1} \boldsymbol\Sigma_0 \right) - k +\ln { | \boldsymbol \Sigma_1 | \over | \boldsymbol\Sigma_0 | }\right\}.

The mutual information of two multivariate normal distribution is a special case of the Kullback–Leibler divergence in which $P$ is the full $k$ dimensional multivariate distribution and $Q$ is the product of the $k\_1$ and $k\_2$ dimensional marginal distributions $X$ and $Y$, such that $k\_1\; +\; k\_2\; =\; k$. The mutual information between $X$ and $Y$ is given by:[https://statproofbook.github.io/P/mvn-mi.html Proof: Mutual information of the multivariate normal distribution]

:$$

I(\boldsymbol{X}, \boldsymbol{Y}) = \frac{1}{2} \ln \left( \frac{\det(\Sigma_X) \det(\Sigma_Y)}{\det(\Sigma)} \right),

where

:$$

\Sigma = \begin{bmatrix}

\Sigma_X & \Sigma_{XY} \\

\Sigma_{XY} & \Sigma_Y

\end{bmatrix}.

If $Q$ is product of $k$ one-dimensional normal distributions, then in the notation of the Kullback–Leibler divergence section of this article, $\backslash boldsymbol\backslash Sigma\_1$ is a diagonal matrix with the diagonal entries of $\backslash boldsymbol\backslash Sigma\_0$, and $\backslash boldsymbol\backslash mu\_1\; =\; \backslash boldsymbol\backslash mu\_0$. The resulting formula for mutual information is:

:$$

I(\boldsymbol{X}) = - { 1 \over 2 } \ln | \boldsymbol \rho_0 |,

where $\backslash boldsymbol\; \backslash rho\_0$ is the correlation matrix constructed from $\backslash boldsymbol\; \backslash Sigma\_0$.{{Cite book |last=MacKay |first=David J. C. |title=Information Theory, Inference and Learning Algorithms |date=2003-10-06 |publisher=Cambridge University Press |isbn=978-0-521-64298-9 |edition=Illustrated |location=Cambridge |language=en}}

In the bivariate case the expression for the mutual information is:

:$$

I(x;y) = - { 1 \over 2 } \ln (1 - \rho^2).

If $X$ and $Y$ are normally distributed and independent, this implies they are "jointly normally distributed", i.e., the pair $(X,Y)$ must have multivariate normal distribution. However, a pair of jointly normally distributed variables need not be independent (would only be so if uncorrelated, $\backslash rho\; =\; 0$ ).

{{See also|normally distributed and uncorrelated does not imply independent}}

The fact that two random variables $X$ and $Y$ both have a normal distribution does not imply that the pair $(X,Y)$ has a joint normal distribution. A simple example is one in which X has a normal distribution with expected value 0 and variance 1, and $Y=X$ if $|X|\; >\; c$ and $Y=-X$ if , where $c\; >\; 0$. There are similar counterexamples for more than two random variables. In general, they sum to a mixture model.{{citation needed|date=August 2020}}

In general, random variables may be uncorrelated but statistically dependent. But if a random vector has a multivariate normal distribution then any two or more of its components that are uncorrelated are independent. This implies that any two or more of its components that are pairwise independent are independent. But, as pointed out just above, it is not true that two random variables that are (separately, marginally) normally distributed and uncorrelated are independent.

If N-dimensional x is partitioned as follows

:$$

\mathbf{x}

=

\begin{bmatrix}

\mathbf{x}_1 \\

\mathbf{x}_2

\end{bmatrix}

\text{ with sizes }\begin{bmatrix} q \times 1 \\ (N-q) \times 1 \end{bmatrix}

and accordingly μ and Σ are partitioned as follows

:$$

\boldsymbol\mu

=

\begin{bmatrix}

\boldsymbol\mu_1 \\

\boldsymbol\mu_2

\end{bmatrix}

\text{ with sizes }\begin{bmatrix} q \times 1 \\ (N-q) \times 1 \end{bmatrix}

:$$

\boldsymbol\Sigma

=

\begin{bmatrix}

\boldsymbol\Sigma_{11} & \boldsymbol\Sigma_{12} \\

\boldsymbol\Sigma_{21} & \boldsymbol\Sigma_{22}

\end{bmatrix}

\text{ with sizes }\begin{bmatrix} q \times q & q \times (N-q) \\ (N-q) \times q & (N-q) \times (N-q) \end{bmatrix}

then the distribution of x₁ conditional on x₂ = a is multivariate normal{{Cite thesis | last1 = Holt | first1 = W. | last2 = Nguyen | first2 = D. | url = https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4494314 | title = Essential Aspects of Bayesian Data Imputation | year = 2023 | ssrn = 4494314 }} {{nowrap|(x₁ {{!}} x₂ {{=}} a) ~ N(μ, Σ)}} where

:$$

\bar{\boldsymbol\mu}

=

\boldsymbol\mu_1 + \boldsymbol\Sigma_{12} \boldsymbol\Sigma_{22}^{-1}

\left(

\mathbf{a} - \boldsymbol\mu_2

\right)

and covariance matrix

:$$

\overline{\boldsymbol\Sigma}

=

\boldsymbol\Sigma_{11} - \boldsymbol\Sigma_{12} \boldsymbol\Sigma_{22}^{-1} \boldsymbol\Sigma_{21}.

{{cite book|last=Eaton|first=Morris L.|title=Multivariate Statistics: a Vector Space Approach|year=1983|publisher=John Wiley and Sons|isbn=978-0-471-02776-8|pages=116–117}}

Here $\backslash boldsymbol\backslash Sigma\_\{22\}^\{-1\}$ is the generalized inverse of $\backslash boldsymbol\backslash Sigma\_\{22\}$. The matrix $\backslash overline\{\backslash boldsymbol\backslash Sigma\}$ is the Schur complement of Σ₂₂ in Σ. That is, the equation above is equivalent to inverting the overall covariance matrix, dropping the rows and columns corresponding to the variables being conditioned upon, and inverting back to get the conditional covariance matrix.

Note that knowing that {{nowrap|x₂ {{=}} a}} alters the variance, though the new variance does not depend on the specific value of a; perhaps more surprisingly, the mean is shifted by $\backslash boldsymbol\backslash Sigma\_\{12\}\; \backslash boldsymbol\backslash Sigma\_\{22\}^\{-1\}\; \backslash left(\backslash mathbf\{a\}\; -\; \backslash boldsymbol\backslash mu\_2\; \backslash right)$; compare this with the situation of not knowing the value of a, in which case x₁ would have distribution

$\backslash mathcal\{N\}\_q\; \backslash left(\backslash boldsymbol\backslash mu\_1,\; \backslash boldsymbol\backslash Sigma\_\{11\}\; \backslash right)$.

An interesting fact derived in order to prove this result, is that the random vectors $\backslash mathbf\{x\}\_2$ and $\backslash mathbf\{y\}\_1=\backslash mathbf\{x\}\_1-\backslash boldsymbol\backslash Sigma\_\{12\}\backslash boldsymbol\backslash Sigma\_\{22\}^\{-1\}\backslash mathbf\{x\}\_2$ are independent.

The matrix Σ₁₂Σ₂₂⁻¹ is known as the matrix of regression coefficients.

In the bivariate case where x is partitioned into $X\_1$ and $X\_2$, the conditional distribution of $X\_1$ given $X\_2$ is{{cite book|last=Jensen|first=J|title=Statistics for Petroleum Engineers and Geoscientists|year=2000|publisher=Elsevier|location=Amsterdam|pages=207|isbn=0-444-50552-0}}

: $X\_1\backslash mid\; X\_2=a\; \backslash \; \backslash sim\backslash \; \backslash mathcal\{N\}\backslash left(\backslash mu\_1+\backslash frac\{\backslash sigma\_1\}\{\backslash sigma\_2\}\backslash rho(\; a\; -\; \backslash mu\_2),\backslash ,\; (1-\backslash rho^2)\backslash sigma\_1^2\backslash right)$

where $\backslash rho\; =\; \backslash frac\{\backslash sigma\_\{12\}\}\{\backslash sigma\_1\backslash sigma\_2\}$ is the correlation coefficient between $X\_1$ and $X\_2$.

:$$

\begin{pmatrix}

X_1 \\

X_2

\end{pmatrix} \sim \mathcal{N} \left( \begin{pmatrix}

\mu_1 \\

\mu_2

\end{pmatrix} , \begin{pmatrix}

\sigma^2_1 & \rho \sigma_1 \sigma_2 \\

\rho \sigma_1 \sigma_2 & \sigma^2_2

\end{pmatrix} \right)

The conditional expectation of X₁ given X₂ is:

: $\backslash operatorname\{E\}(X\_1\; \backslash mid\; X\_2=x\_2)\; =\; \backslash mu\_1\; +\; \backslash rho\; \backslash frac\{\backslash sigma\_1\}\{\backslash sigma\_2\}(x\_2\; -\; \backslash mu\_2)$

Proof: the result is obtained by taking the expectation of the conditional distribution $X\_1\backslash mid\; X\_2$ above.

:$$

\begin{pmatrix}

X_1 \\

X_2

\end{pmatrix} \sim \mathcal{N} \left( \begin{pmatrix}

0 \\

0

\end{pmatrix} , \begin{pmatrix}

1 & \rho \\

\rho & 1

\end{pmatrix} \right)

The conditional expectation of X₁ given X₂ is

: $\backslash operatorname\{E\}(X\_1\; \backslash mid\; X\_2=x\_2)=\; \backslash rho\; x\_2$

and the conditional variance is

: $\backslash operatorname\{var\}(X\_1\; \backslash mid\; X\_2\; =\; x\_2)\; =\; 1-\backslash rho^2;$

thus the conditional variance does not depend on x₂.

The conditional expectation of X₁ given that X₂ is smaller/bigger than z is:{{cite book|last=Maddala|first=G. S.|title=Limited Dependent and Qualitative Variables in Econometrics|year=1983|publisher=Cambridge University Press|isbn=0-521-33825-5}}{{rp|367}}

:$$

\operatorname{E}(X_1 \mid X_2 < z) = -\rho { \varphi(z) \over \Phi(z) } ,

:$$

\operatorname{E}(X_1 \mid X_2 > z) = \rho { \varphi(z) \over (1- \Phi(z)) } ,

where the final ratio here is called the inverse Mills ratio.

Proof: the last two results are obtained using the result $\backslash operatorname\{E\}(X\_1\; \backslash mid\; X\_2=x\_2)=\; \backslash rho\; x\_2$, so that

:$$

\operatorname{E}(X_1 \mid X_2 < z) = \rho E(X_2 \mid X_2 < z) and then using the properties of the expectation of a truncated normal distribution.

To obtain the marginal distribution over a subset of multivariate normal random variables, one only needs to drop the irrelevant variables (the variables that one wants to marginalize out) from the mean vector and the covariance matrix. The proof for this follows from the definitions of multivariate normal distributions and linear algebra.An algebraic computation of the marginal distribution is shown here http://fourier.eng.hmc.edu/e161/lectures/gaussianprocess/node7.html {{Webarchive|url=https://web.archive.org/web/20100117200722/http://fourier.eng.hmc.edu/e161/lectures/gaussianprocess/node7.html |date=2010-01-17 }}. A much shorter proof is outlined here https://math.stackexchange.com/a/3832137

Example

Let {{nowrap|X {{=}} [X₁, X₂, X₃]}} be multivariate normal random variables with mean vector {{nowrap|μ {{=}} [μ₁, μ₂, μ₃]}} and covariance matrix Σ (standard parametrization for multivariate normal distributions). Then the joint distribution of {{nowrap|{{prime|'X}} {{=}} [X₁, X₃]}} is multivariate normal with mean vector {{nowrap|{{prime|'μ}} {{=}} [μ₁, μ₃]}} and covariance matrix

$\backslash boldsymbol\backslash Sigma\text{'}\; =$

\begin{bmatrix}

\boldsymbol\Sigma_{11} & \boldsymbol\Sigma_{13} \\

\boldsymbol\Sigma_{31} & \boldsymbol\Sigma_{33}

\end{bmatrix}

.

If {{nowrap|Y {{=}} c + BX}} is an affine transformation of $\backslash mathbf\{X\}\backslash \; \backslash sim\; \backslash mathcal\{N\}(\backslash boldsymbol\backslash mu,\; \backslash boldsymbol\backslash Sigma),$ where c is an $M\; \backslash times\; 1$ vector of constants and B is a constant $M\; \backslash times\; N$ matrix, then Y has a multivariate normal distribution with expected value {{nowrap|c + Bμ}} and variance BΣB^T i.e., $\backslash mathbf\{Y\}\; \backslash sim\; \backslash mathcal\{N\}\; \backslash left(\backslash mathbf\{c\}\; +\; \backslash mathbf\{B\}\; \backslash boldsymbol\backslash mu,\; \backslash mathbf\{B\}\; \backslash boldsymbol\backslash Sigma\; \backslash mathbf\{B\}^\{\backslash rm\; T\}\backslash right)$. In particular, any subset of the X_i has a marginal distribution that is also multivariate normal.

To see this, consider the following example: to extract the subset (X₁, X₂, X₄)^T, use

:$$

\mathbf{B}

=

\begin{bmatrix}

1 & 0 & 0 & 0 & 0 & \ldots & 0 \\

0 & 1 & 0 & 0 & 0 & \ldots & 0 \\

0 & 0 & 0 & 1 & 0 & \ldots & 0

\end{bmatrix}

which extracts the desired elements directly.

Another corollary is that the distribution of {{nowrap|Z {{=}} b · X}}, where b is a constant vector with the same number of elements as X and the dot indicates the dot product, is univariate Gaussian with $Z\backslash sim\backslash mathcal\{N\}\backslash left(\backslash mathbf\{b\}\backslash cdot\backslash boldsymbol\backslash mu,\; \backslash mathbf\{b\}^\{\backslash rm\; T\}\backslash boldsymbol\backslash Sigma\; \backslash mathbf\{b\}\backslash right)$. This result follows by using

:$$

\mathbf{B}=\begin{bmatrix}

b_1 & b_2 & \ldots & b_n

\end{bmatrix} = \mathbf{b}^{\rm T}.

Observe how the positive-definiteness of Σ implies that the variance of the dot product must be positive.

An affine transformation of X such as 2X is not the same as the sum of two independent realisations of X.

{{see also|Confidence region}}

The equidensity contours of a non-singular multivariate normal distribution are ellipsoids (i.e. affine transformations of hyperspheres) centered at the mean.{{cite web|author=Nikolaus Hansen|title=The CMA Evolution Strategy: A Tutorial|url=http://www.lri.fr/~hansen/cmatutorial.pdf|access-date=2012-01-07|archive-url=https://web.archive.org/web/20100331114258/http://www.lri.fr/~hansen/cmatutorial.pdf|archive-date=2010-03-31|url-status=dead|bibcode=2016arXiv160400772H|year=2016|arxiv=1604.00772}} Hence the multivariate normal distribution is an example of the class of elliptical distributions. The directions of the principal axes of the ellipsoids are given by the eigenvectors of the covariance matrix $\backslash boldsymbol\backslash Sigma$. The squared relative lengths of the principal axes are given by the corresponding eigenvalues.

If {{nowrap|Σ {{=}} UΛU^T {{=}} UΛ^1/2(UΛ^1/2)^T}} is an eigendecomposition where the columns of U are unit eigenvectors and Λ is a diagonal matrix of the eigenvalues, then we have

::$\backslash mathbf\{X\}\backslash \; \backslash sim\; \backslash mathcal\{N\}(\backslash boldsymbol\backslash mu,\; \backslash boldsymbol\backslash Sigma)\; \backslash iff\; \backslash mathbf\{X\}\backslash \; \backslash sim\; \backslash boldsymbol\backslash mu+\backslash mathbf\{U\}\backslash boldsymbol\backslash Lambda^\{1/2\}\backslash mathcal\{N\}(0,\; \backslash mathbf\{I\})\; \backslash iff\; \backslash mathbf\{X\}\backslash \; \backslash sim\; \backslash boldsymbol\backslash mu+\backslash mathbf\{U\}\backslash mathcal\{N\}(0,\; \backslash boldsymbol\backslash Lambda).$

Moreover, U can be chosen to be a rotation matrix, as inverting an axis does not have any effect on N(0, Λ), but inverting a column changes the sign of U's determinant. The distribution N(μ, Σ) is in effect N(0, I) scaled by Λ^1/2, rotated by U and translated by μ.

Conversely, any choice of μ, full rank matrix U, and positive diagonal entries Λ_i yields a non-singular multivariate normal distribution. If any Λ_i is zero and U is square, the resulting covariance matrix UΛU^T is singular. Geometrically this means that every contour ellipsoid is infinitely thin and has zero volume in n-dimensional space, as at least one of the principal axes has length of zero; this is the degenerate case.

"The radius around the true mean in a bivariate normal random variable, re-written in polar coordinates (radius and angle), follows a Hoyt distribution."{{cite web |title=The Hoyt Distribution (Documentation for R package 'shotGroups' version 0.6.2) |author=Daniel Wollschlaeger |url=http://finzi.psych.upenn.edu/usr/share/doc/library/shotGroups/html/hoyt.html }}{{dead link|date=December 2017 |bot=InternetArchiveBot |fix-attempted=yes }}

In one dimension the probability of finding a sample of the normal distribution in the interval $\backslash mu\backslash pm\; \backslash sigma$ is approximately 68.27%, but in higher dimensions the probability of finding a sample in the region of the standard deviation ellipse is lower.{{Cite journal|last1=Wang|first1=Bin|last2=Shi|first2=Wenzhong|last3=Miao|first3=Zelang|date=2015-03-13|editor-last=Rocchini|editor-first=Duccio|title=Confidence Analysis of Standard Deviational Ellipse and Its Extension into Higher Dimensional Euclidean Space|journal=PLOS ONE|language=en|volume=10|issue=3|pages=e0118537|doi=10.1371/journal.pone.0118537|issn=1932-6203|pmc=4358977|pmid=25769048|bibcode=2015PLoSO..1018537W|doi-access=free}}

{{further|Estimation of covariance matrices}}

The derivation of the maximum-likelihood estimator of the covariance matrix of a multivariate normal distribution is straightforward.

In short, the probability density function (pdf) of a multivariate normal is

:$f(\backslash mathbf\{x\})=\; \backslash frac\{1\}\{\backslash sqrt\; \{\; (2\backslash pi)^k|\backslash boldsymbol\; \backslash Sigma|\; \}\; \}\; \backslash exp\backslash left(-\{1\; \backslash over\; 2\}\; (\backslash mathbf\{x\}-\backslash boldsymbol\backslash mu)^\{\backslash rm\; T\}\; \backslash boldsymbol\backslash Sigma^\{-1\}\; (\{\backslash mathbf\; x\}-\backslash boldsymbol\backslash mu)\backslash right)$

and the ML estimator of the covariance matrix from a sample of n observations is {{Cite thesis | last1 = Holt | first1 = W. | last2 = Nguyen | first2 = D. | url = https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4494314 | title = Introduction to Bayesian Data Imputation | year = 2023 | ssrn = 4494314 }}

:$\backslash widehat\{\backslash boldsymbol\backslash Sigma\}\; =\; \{1\; \backslash over\; n\}\backslash sum\_\{i=1\}^n\; (\{\backslash mathbf\; x\}\_i-\backslash overline\{\backslash mathbf\; x\})(\{\backslash mathbf\; x\}\_i-\backslash overline\{\backslash mathbf\; x\})^\backslash mathrm\{T\}$

which is simply the sample covariance matrix. This is a biased estimator whose expectation is

:$E\backslash left[\backslash widehat\{\backslash boldsymbol\backslash Sigma\}\backslash right]\; =\; \backslash frac\{n-1\}\{n\}\; \backslash boldsymbol\backslash Sigma.$

An unbiased sample covariance is

:$\backslash widehat\{\backslash boldsymbol\backslash Sigma\}\; =\; \backslash frac1\{n-1\}\backslash sum\_\{i=1\}^n\; (\backslash mathbf\{x\}\_i-\backslash overline\{\backslash mathbf\{x\}\})(\backslash mathbf\{x\}\_i-\backslash overline\{\backslash mathbf\{x\}\})^\{\backslash rm\; T\}$

= \frac1{n-1} \left[X'\left(I - \frac{1}{n} \cdot J\right) X\right] (matrix form; $I$ is the $K\backslash times\; K$ identity matrix, J is a $K\; \backslash times\; K$ matrix of ones; the term in parentheses is thus the $K\; \backslash times\; K$ centering matrix)

The Fisher information matrix for estimating the parameters of a multivariate normal distribution has a closed form expression. This can be used, for example, to compute the Cramér–Rao bound for parameter estimation in this setting. See Fisher information for more details.

In Bayesian statistics, the conjugate prior of the mean vector is another multivariate normal distribution, and the conjugate prior of the covariance matrix is an inverse-Wishart distribution $\backslash mathcal\{W\}^\{-1\}$ . Suppose then that n observations have been made

:$\backslash mathbf\{X\}\; =\; \backslash \{\backslash mathbf\{x\}\_1,\backslash dots,\backslash mathbf\{x\}\_n\backslash \}\; \backslash sim\; \backslash mathcal\{N\}(\backslash boldsymbol\backslash mu,\backslash boldsymbol\backslash Sigma)$

and that a conjugate prior has been assigned, where

:$p(\backslash boldsymbol\backslash mu,\backslash boldsymbol\backslash Sigma)=p(\backslash boldsymbol\backslash mu\backslash mid\backslash boldsymbol\backslash Sigma)\backslash \; p(\backslash boldsymbol\backslash Sigma),$

where

:$p(\backslash boldsymbol\backslash mu\backslash mid\backslash boldsymbol\backslash Sigma)\; \backslash sim\backslash mathcal\{N\}(\backslash boldsymbol\backslash mu\_0,m^\{-1\}\backslash boldsymbol\backslash Sigma)\; ,$

and

:$p(\backslash boldsymbol\backslash Sigma)\; \backslash sim\; \backslash mathcal\{W\}^\{-1\}(\backslash boldsymbol\backslash Psi,n\_0).$

Then

:$$

\begin{array}{rcl}

p(\boldsymbol\mu\mid\boldsymbol\Sigma,\mathbf{X}) & \sim & \mathcal{N}\left(\frac{n\bar{\mathbf{x}} + m\boldsymbol\mu_0}{n+m},\frac{1}{n+m}\boldsymbol\Sigma\right),\\

p(\boldsymbol\Sigma\mid\mathbf{X}) & \sim & \mathcal{W}^{-1}\left(\boldsymbol\Psi+n\mathbf{S}+\frac{nm}{n+m}(\bar{\mathbf{x}}-\boldsymbol\mu_0)(\bar{\mathbf{x}}-\boldsymbol\mu_0)', n+n_0\right),

\end{array}

where

:$$

\begin{align}

\bar{\mathbf{x}} & = \frac{1}{n}\sum_{i=1}^{n} \mathbf{x}_i ,\\

\mathbf{S} & = \frac{1}{n}\sum_{i=1}^{n} (\mathbf{x}_i - \bar{\mathbf{x}})(\mathbf{x}_i - \bar{\mathbf{x}})' .

\end{align}

Multivariate normality tests check a given set of data for similarity to the multivariate normal distribution. The null hypothesis is that the data set is similar to the normal distribution, therefore a sufficiently small p-value indicates non-normal data. Multivariate normality tests include the Cox–Small test{{Cite journal | last1 = Cox | first1 = D. R. | last2 = Small | first2 = N. J. H. | doi = 10.1093/biomet/65.2.263 | title = Testing multivariate normality | journal = Biometrika | volume = 65 | issue = 2 | pages = 263 | year = 1978 }}

and Smith and Jain's adaptation{{Cite journal | last1 = Smith | first1 = S. P. | last2 = Jain | first2 = A. K. | doi = 10.1109/34.6789 | title = A test to determine the multivariate normality of a data set | journal = IEEE Transactions on Pattern Analysis and Machine Intelligence | volume = 10 | issue = 5 | pages = 757 | year = 1988 }} of the Friedman–Rafsky test created by Larry Rafsky and Jerome Friedman.{{Cite journal | last1 = Friedman | first1 = J. H. | last2 = Rafsky | first2 = L. C. | doi = 10.1214/aos/1176344722 | title = Multivariate Generalizations of the Wald–Wolfowitz and Smirnov Two-Sample Tests | journal = The Annals of Statistics | volume = 7 | issue = 4 | pages = 697 | year = 1979 | doi-access = free }}

Mardia's test is based on multivariate extensions of skewness and kurtosis measures. For a sample {x₁, ..., x_n} of k-dimensional vectors we compute

: $\backslash begin\{align\}$

& \widehat{\boldsymbol\Sigma} = {1 \over n} \sum_{j=1}^n \left(\mathbf{x}_j - \bar{\mathbf{x}}\right)\left(\mathbf{x}_j - \bar{\mathbf{x}}\right)^\mathrm{T} \\

& A = {1 \over 6n} \sum_{i=1}^n \sum_{j=1}^n \left[ (\mathbf{x}_i - \bar{\mathbf{x}})^\mathrm{T}\;\widehat{\boldsymbol\Sigma}^{-1} (\mathbf{x}_j - \bar{\mathbf{x}}) \right]^3 \\

& B = \sqrt{\frac{n}{8k(k+2)}}\left\{{1 \over n} \sum_{i=1}^n \left[ (\mathbf{x}_i - \bar{\mathbf{x}})^\mathrm{T}\;\widehat{\boldsymbol\Sigma}^{-1} (\mathbf{x}_i - \bar{\mathbf{x}}) \right]^2 - k(k+2) \right\}

\end{align}

Under the null hypothesis of multivariate normality, the statistic A will have approximately a chi-squared distribution with {{nowrap|{{frac2|1|6}}⋅k(k + 1)(k + 2)}} degrees of freedom, and B will be approximately standard normal N(0,1).

Mardia's kurtosis statistic is skewed and converges very slowly to the limiting normal distribution. For medium size samples , the parameters of the asymptotic distribution of the kurtosis statistic are modifiedRencher (1995), pages 112–113. For small sample tests () empirical critical values are used. Tables of critical values for both statistics are given by RencherRencher (1995), pages 493–495. for k = 2, 3, 4.

Mardia's tests are affine invariant but not consistent. For example, the multivariate skewness test is not consistent against

symmetric non-normal alternatives.{{Cite journal | last1 = Baringhaus | first1 = L. | last2 = Henze | first2 = N. | doi = 10.1016/0047-259X(91)90031-V | title = Limit distributions for measures of multivariate skewness and kurtosis based on projections | journal = Journal of Multivariate Analysis | volume = 38 | pages = 51–69 | year = 1991 | doi-access = free }}

The BHEP test computes the norm of the difference between the empirical characteristic function and the theoretical characteristic function of the normal distribution. Calculation of the norm is performed in the L²(μ) space of square-integrable functions with respect to the Gaussian weighting function $\backslash mu\_\backslash beta(\backslash mathbf\{t\})\; =\; (2\backslash pi\backslash beta^2)^\{-k/2\}\; e^\{-|\backslash mathbf\{t\}|^2/(2\backslash beta^2)\}$. The test statistic is

: $\backslash begin\{align\}$

T_\beta &= \int_{\mathbb{R}^k} \left| {1 \over n} \sum_{j=1}^n e^{i\mathbf{t}^\mathrm{T}\widehat{\boldsymbol\Sigma}^{-1/2}(\mathbf{x}_j - \bar{\mathbf{x})}} - e^{-|\mathbf{t}|^2/2} \right|^2 \; \boldsymbol\mu_\beta(\mathbf{t}) \, d\mathbf{t} \\

&= {1 \over n^2} \sum_{i,j=1}^n e^{-{\beta^2 \over 2}(\mathbf{x}_i-\mathbf{x}_j)^\mathrm{T}\widehat{\boldsymbol\Sigma}^{-1}(\mathbf{x}_i-\mathbf{x}_j)} - \frac{2}{n(1 + \beta^2)^{k/2}}\sum_{i=1}^n e^{ -\frac{\beta^2}{2(1+\beta^2)} (\mathbf{x}_i-\bar{\mathbf{x}})^\mathrm{T}\widehat{\boldsymbol\Sigma}^{-1}(\mathbf{x}_i-\bar{\mathbf{x}})} + \frac{1}{(1 + 2\beta^2)^{k/2}}

\end{align}

The limiting distribution of this test statistic is a weighted sum of chi-squared random variables.

A detailed survey of these and other test procedures is available.

[[File:Classification of several multivariate normals.png|right|thumb|600px|Left: Classification of seven multivariate normal classes. Coloured ellipses are 1 sd error ellipses. Black marks the boundaries between the classification regions. $p\_e$ is the probability of total classification error. Right: the error matrix. $p\_\{ij\}$ is the probability of classifying a sample from normal $i$ as $j$. These are computed by the numerical method of ray-tracing ([https://www.mathworks.com/matlabcentral/fileexchange/84973-integrate-and-classify-normal-distributions Matlab code]).

]]

Suppose that observations (which are vectors) are presumed to come from one of several multivariate normal distributions, with known means and covariances. Then any given observation can be assigned to the distribution from which it has the highest probability of arising. This classification procedure is called Gaussian discriminant analysis.

The classification performance, i.e. probabilities of the different classification outcomes, and the overall classification error, can be computed by the numerical method of ray-tracing ([https://www.mathworks.com/matlabcentral/fileexchange/84973-integrate-and-classify-normal-distributions Matlab code]).

A widely used method for drawing (sampling) a random vector x from the N-dimensional multivariate normal distribution with mean vector μ and covariance matrix Σ works as follows:

1. Find any real matrix A such that {{nowrap|AA^T {{=}} Σ}}. When Σ is positive-definite, the Cholesky decomposition is typically used because it is widely available, computationally efficient, and well known. If a rank-revealing (pivoted) Cholesky decomposition such as LAPACK's dpstrf() is available, it can be used in the general positive-semidefinite case as well. A slower general alternative is to use the matrix A = UΛ^1/2 obtained from a spectral decomposition Σ = UΛU⁻¹ of Σ.
2. Let {{nowrap|z {{=}} (z₁, ..., z_N)^T}} be a vector whose components are N independent standard normal variates (which can be generated, for example, by using the Box–Muller transform).
3. Let x be {{nowrap|μ + Az}}. This has the desired distribution due to the affine transformation property.

• Chi distribution, the pdf of the 2-norm (Euclidean norm or vector length) of a multivariate normally distributed vector (uncorrelated and zero centered).
• Rayleigh distribution, the pdf of the vector length of a bivariate normally distributed vector (uncorrelated and zero centered)
• Rice distribution, the pdf of the vector length of a bivariate normally distributed vector (uncorrelated and non-centered)
• Hoyt distribution, the pdf of the vector length of a bivariate normally distributed vector (correlated and centered)
• Complex normal distribution, an application of bivariate normal distribution
• Copula, for the definition of the Gaussian or normal copula model.
• Multivariate t-distribution, which is another widely used spherically symmetric multivariate distribution.
• Multivariate stable distribution extension of the multivariate normal distribution, when the index (exponent in the characteristic function) is between zero and two.
• Mahalanobis distance
• Wishart distribution
• Matrix normal distribution

{{Reflist|refs=

{{cite journal

| last = Siotani |first=Minoru

| title = Tolerance regions for a multivariate normal population

| journal = Annals of the Institute of Statistical Mathematics

| year = 1964

| volume = 16

| number = 1

| pages = 135–153

| doi = 10.1007/BF02868568

|s2cid=123269490

| url = http://www.ism.ac.jp/editsec/aism/pdf/016_1_0135.pdf

}}

{{cite journal

| last = Mardia | first = K. V.

| year = 1970

| title = Measures of multivariate skewness and kurtosis with applications

| journal = Biometrika

| volume = 57 | issue = 3 | pages = 519–530

| doi = 10.1093/biomet/57.3.519

}}

{{cite journal

| last1 = Baringhaus | first1 = L.

| last2 = Henze | first2 = N.

| year = 1988

| title = A consistent test for multivariate normality based on the empirical characteristic function

| journal = Metrika

| volume = 35 | issue = 1 | pages = 339–348

| doi = 10.1007/BF02613322

| s2cid = 122362448

}}

{{cite journal

| last = Henze | first = Norbert

| year = 2002

| title = Invariant tests for multivariate normality: a critical review

| journal = Statistical Papers

| volume = 43 | issue = 4 | pages = 467–506

| doi = 10.1007/s00362-002-0119-6

| s2cid = 122934510

}}

{{cite journal

| last1 = Hamedani | first1 = G. G.

| last2 = Tata | first2 = M. N.

| year = 1975

| title = On the determination of the bivariate normal distribution from distributions of linear combinations of the variables

| journal = The American Mathematical Monthly

| volume = 82 | issue = 9 | pages = 913–915

| doi = 10.2307/2318494

| jstor = 2318494

}}

{{cite web |last=Wyatt |first=John

| title=Linear least mean-squared error estimation

| url=http://web.mit.edu/6.041/www/LECTURE/lec22.pdf

| work=Lecture notes course on applied probability

| date=November 26, 2008

| archive-date=October 10, 2015

| archive-url=https://web.archive.org/web/20151010114443/http://web.mit.edu/6.041/www/LECTURE/lec22.pdf

| access-date=23 January 2012 }}

{{cite book

| last = Rao |first=C. R. |author-link=C. R. Rao

| title = Linear Statistical Inference and Its Applications

| year = 1973

| publisher = Wiley

| location = New York

| pages = 527–528

| isbn = 0-471-70823-2

}}

{{cite book

| last = Gentle |first=J. E.

| title = Computational Statistics

| year = 2009

| publisher = Springer

| location = New York

| pages = 315–316

| doi = 10.1007/978-0-387-98144-4

| series = Statistics and Computing

| isbn = 978-0-387-98143-7

| url = https://cds.cern.ch/record/1639470

}}

{{cite journal

| title=The General Projected Normal Distribution of Arbitrary Dimension: Modeling and Bayesian Inference

| year=2017

| journal=Bayesian Analysis

| volume=12

| pages=113–133

| issue=1

| last1=Hernandez-Stumpfhauser

| first1=Daniel

| last2=Breidt

| first2=F. Jay

| last3=van der Woerd

| first3=Mark J.

| doi=10.1214/15-BA989

| doi-access=free

}}

}}

{{refbegin}}

• {{cite book

| author = Rencher, A.C.

| title = Methods of Multivariate Analysis

| year = 1995

| publisher = Wiley

| location = New York

}}

• {{cite book |first=Y. L. |last=Tong |title=The multivariate normal distribution |year=1990 |isbn=978-1-4613-9657-4 |series=Springer Series in Statistics |location=New York |publisher=Springer-Verlag |doi=10.1007/978-1-4613-9655-0|s2cid=120348131 }}

{{refend}}

{{ProbDistributions|multivariate|state=collapsed}}

{{statistics|analysis|state=collapsed}}

{{DEFAULTSORT:Multivariate Normal Distribution}}

Category:Continuous distributions

Category:Multivariate continuous distributions

Category:Normal distribution

Category:Exponential family distributions

Category:Stable distributions