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Inprobability andstatistics, amultivariate random variable orrandom vector is a list orvector of mathematicalvariables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its value. The individual variables in a random vector are grouped together because they are all part of a single mathematical system — often they represent different properties of an individualstatistical unit. For example, while a given person has a specific age, height and weight, the representation of these features ofan unspecified person from within a group would be a random vector. Normally each element of a random vector is areal number.

Random vectors are often used as the underlying implementation of various types of aggregaterandom variables, e.g. arandom matrix,random tree,random sequence,stochastic process, etc.

Formally, a multivariate random variable is acolumn vector${\displaystyle \mathbf{X}=(X_1,\dots ,X_n{)}^{\mathsf{T}}}$ (or itstranspose, which is arow vector) whose components arerandom variables on theprobability space${\displaystyle (\mathrm{\Omega },\mathcal{F},P)}$, where${\displaystyle \mathrm{\Omega }}$ is thesample space,${\displaystyle \mathcal{F}}$ is thesigma-algebra (the collection of all events), and${\displaystyle P}$ is theprobability measure (a function returning each event'sprobability).

Every random vector gives rise to a probability measure on${\displaystyle {\mathbb{R}}^n}$ with theBorel algebra as the underlying sigma-algebra. This measure is also known as thejoint probability distribution, the joint distribution, or the multivariate distribution of the random vector.

Thedistributions of each of the component random variables${\displaystyle X_i}$ are calledmarginal distributions. Theconditional probability distribution of${\displaystyle X_i}$ given${\displaystyle X_j}$ is the probability distribution of${\displaystyle X_i}$ when${\displaystyle X_j}$ is known to be a particular value.

Thecumulative distribution function${\displaystyle F_{\mathbf{X}}:{\mathbb{R}}^n\mapsto [0,1]}$ of a random vector${\displaystyle \mathbf{X}=(X_1,\dots ,X_n{)}^{\mathsf{T}}}$ is defined as^[1]: p.15

${\displaystyle F_{\mathbf{X}}(\mathbf{x})=\mathrm{P}(X_1\le x_1,\dots ,X_n\le x_n)}$

Eq.1

where${\displaystyle \mathbf{x}=(x_1,\dots ,x_n{)}^{\mathsf{T}}}$.

Random vectors can be subjected to the same kinds ofalgebraic operations as can non-random vectors: addition, subtraction, multiplication by ascalar, and the taking ofinner products.

Similarly, a new random vector${\displaystyle \mathbf{Y}}$ can be defined by applying anaffine transformation${\displaystyle g:{\mathbb{R}}^n\to {\mathbb{R}}^n}$ to a random vector${\displaystyle \mathbf{X}}$:

${\displaystyle \mathbf{Y}=\mathbf{A}\mathbf{X}+b}$, where${\displaystyle \mathbf{A}}$ is an${\displaystyle n\times n}$ matrix and${\displaystyle b}$ is an${\displaystyle n\times 1}$ column vector.

If${\displaystyle \mathbf{A}}$ is aninvertible matrix and${\displaystyle \mathbf{X}}$ has a probability density function${\displaystyle f_{\mathbf{X}}}$, then the probability density of${\displaystyle \mathbf{Y}}$ is

${\displaystyle f_{\mathbf{Y}}(y)=\frac{f_{\mathbf{X}}({\mathbf{A}}^{-1}(y-b))}{|det\mathbf{A}|}}$.

More generally we can study invertible mappings of random vectors.^{[2]: p.284–285}

Let${\displaystyle g}$ be a one-to-one mapping from an open subset${\displaystyle \mathcal{D}}$ of${\displaystyle {\mathbb{R}}^n}$ onto a subset${\displaystyle \mathcal{R}}$ of${\displaystyle {\mathbb{R}}^n}$, let${\displaystyle g}$ have continuous partial derivatives in${\displaystyle \mathcal{D}}$ and let theJacobian determinant${\displaystyle det\left(\frac{\mathrm{\partial }\mathbf{y}}{\mathrm{\partial }\mathbf{x}}\right)}$ of${\displaystyle g}$ be zero at no point of${\displaystyle \mathcal{D}}$. Assume that the real random vector${\displaystyle \mathbf{X}}$ has a probability density function${\displaystyle f_{\mathbf{X}}(\mathbf{x})}$ and satisfies${\displaystyle P(\mathbf{X}\in \mathcal{D})=1}$. Then the random vector${\displaystyle \mathbf{Y}=g(\mathbf{X})}$ is of probability density

${\displaystyle {f_{\mathbf{Y}}(\mathbf{y})=\frac{f_{\mathbf{X}}(\mathbf{x})}{\left|det\left(\frac{\mathrm{\partial }\mathbf{y}}{\mathrm{\partial }\mathbf{x}}\right)\right|}|}_{\mathbf{x}=g^{-1}(\mathbf{y})}\mathbf{1}(\mathbf{y}\in R_{\mathbf{Y}})}$

where${\displaystyle \mathbf{1}}$ denotes theindicator function and set${\displaystyle R_{\mathbf{Y}}=\{\mathbf{y}=g(\mathbf{x}):f_{\mathbf{X}}(\mathbf{x})>0\}\subseteq \mathcal{R}}$ denotes support of${\displaystyle \mathbf{Y}}$.

Theexpected value or mean of a random vector${\displaystyle \mathbf{X}}$ is a fixed vector${\displaystyle \mathrm{E}[\mathbf{X}]}$ whose elements are the expected values of the respective random variables.^[3]: p.333

${\displaystyle \mathrm{E}[\mathbf{X}]=(\mathrm{E}[X_1],...,\mathrm{E}[X_n]{)}^{\mathrm{T}}}$

Eq.2

Thecovariance matrix (also calledsecond central moment or variance-covariance matrix) of an${\displaystyle n\times 1}$ random vector is an${\displaystyle n\times n}$matrix whose (i,j)^th element is thecovariance between thei ^th and thej ^th random variables. The covariance matrix is the expected value, element by element, of the${\displaystyle n\times n}$ matrixcomputed as${\displaystyle [\mathbf{X}-\mathrm{E}[\mathbf{X}]][\mathbf{X}-\mathrm{E}[\mathbf{X}]{]}^T}$, where the superscript T refers to the transpose of the indicated vector:^{[2]: p. 464 [3]: p.335}

${\displaystyle {\mathrm{K}}_{\mathbf{X}\mathbf{X}}=\mathrm{Var}[\mathbf{X}]=\mathrm{E}[(\mathbf{X}-\mathrm{E}[\mathbf{X}])(\mathbf{X}-\mathrm{E}[\mathbf{X}]{)}^T]=\mathrm{E}[\mathbf{X}{\mathbf{X}}^T]-\mathrm{E}[\mathbf{X}]\mathrm{E}[\mathbf{X}{]}^T}$

Eq.3

By extension, thecross-covariance matrix between two random vectors${\displaystyle \mathbf{X}}$ and${\displaystyle \mathbf{Y}}$ (${\displaystyle \mathbf{X}}$ having${\displaystyle n}$ elements and${\displaystyle \mathbf{Y}}$ having${\displaystyle p}$ elements) is the${\displaystyle n\times p}$ matrix^[3]: p.336

${\displaystyle {\mathrm{K}}_{\mathbf{X}\mathbf{Y}}=\mathrm{Cov}[\mathbf{X},\mathbf{Y}]=\mathrm{E}[(\mathbf{X}-\mathrm{E}[\mathbf{X}])(\mathbf{Y}-\mathrm{E}[\mathbf{Y}]{)}^T]=\mathrm{E}[\mathbf{X}{\mathbf{Y}}^T]-\mathrm{E}[\mathbf{X}]\mathrm{E}[\mathbf{Y}{]}^T}$

Eq.4

where again the matrix expectation is taken element-by-element in the matrix. Here the (i,j)^th element is the covariance between thei ^th element of${\displaystyle \mathbf{X}}$ and thej ^th element of${\displaystyle \mathbf{Y}}$.

The covariance matrix is asymmetric matrix, i.e.^{[2]: p. 466}

${\displaystyle {\mathrm{K}}_{\mathbf{X}\mathbf{X}}^T={\mathrm{K}}_{\mathbf{X}\mathbf{X}}}$.

The covariance matrix is apositive semidefinite matrix, i.e.^{[2]: p. 465}

${\displaystyle {\mathbf{a}}^T{\mathrm{K}}_{\mathbf{X}\mathbf{X}}\mathbf{a}\ge 0\text{for all }\mathbf{a}\in {\mathbb{R}}^n}$.

The cross-covariance matrix${\displaystyle \mathrm{Cov}[\mathbf{Y},\mathbf{X}]}$ is simply the transpose of the matrix${\displaystyle \mathrm{Cov}[\mathbf{X},\mathbf{Y}]}$, i.e.

${\displaystyle {\mathrm{K}}_{\mathbf{Y}\mathbf{X}}={\mathrm{K}}_{\mathbf{X}\mathbf{Y}}^T}$.

Two random vectors${\displaystyle \mathbf{X}=(X_1,...,X_m{)}^T}$ and${\displaystyle \mathbf{Y}=(Y_1,...,Y_n{)}^T}$ are calleduncorrelated if

${\displaystyle \mathrm{E}[\mathbf{X}{\mathbf{Y}}^T]=\mathrm{E}[\mathbf{X}]\mathrm{E}[\mathbf{Y}{]}^T}$.

They are uncorrelatedif and only if their cross-covariance matrix${\displaystyle {\mathrm{K}}_{\mathbf{X}\mathbf{Y}}}$ is zero.^[3]: p.337

Thecorrelation matrix (also calledsecond moment) of an${\displaystyle n\times 1}$ random vector is an${\displaystyle n\times n}$ matrix whose (i,j)^th element is the correlation between thei ^th and thej ^th random variables. The correlation matrix is the expected value, element by element, of the${\displaystyle n\times n}$ matrix computed as${\displaystyle \mathbf{X}{\mathbf{X}}^T}$, where the superscript T refers to the transpose of the indicated vector:^{[4]: p.190 [3]: p.334}

${\displaystyle {\mathrm{R}}_{\mathbf{X}\mathbf{X}}=\mathrm{E}[\mathbf{X}{\mathbf{X}}^{\mathrm{T}}]}$

Eq.5

By extension, thecross-correlation matrix between two random vectors${\displaystyle \mathbf{X}}$ and${\displaystyle \mathbf{Y}}$ (${\displaystyle \mathbf{X}}$ having${\displaystyle n}$ elements and${\displaystyle \mathbf{Y}}$ having${\displaystyle p}$ elements) is the${\displaystyle n\times p}$ matrix

${\displaystyle {\mathrm{R}}_{\mathbf{X}\mathbf{Y}}=\mathrm{E}[\mathbf{X}{\mathbf{Y}}^T]}$

Eq.6

The correlation matrix is related to the covariance matrix by

${\displaystyle {\mathrm{R}}_{\mathbf{X}\mathbf{X}}={\mathrm{K}}_{\mathbf{X}\mathbf{X}}+\mathrm{E}[\mathbf{X}]\mathrm{E}[\mathbf{X}{]}^T}$.

Similarly for the cross-correlation matrix and the cross-covariance matrix:

${\displaystyle {\mathrm{R}}_{\mathbf{X}\mathbf{Y}}={\mathrm{K}}_{\mathbf{X}\mathbf{Y}}+\mathrm{E}[\mathbf{X}]\mathrm{E}[\mathbf{Y}{]}^T}$

Two random vectors of the same size${\displaystyle \mathbf{X}=(X_1,...,X_n{)}^T}$ and${\displaystyle \mathbf{Y}=(Y_1,...,Y_n{)}^T}$ are calledorthogonal if

${\displaystyle \mathrm{E}[{\mathbf{X}}^T\mathbf{Y}]=0}$.

Two random vectors${\displaystyle \mathbf{X}}$ and${\displaystyle \mathbf{Y}}$ are calledindependent if for all${\displaystyle \mathbf{x}}$ and${\displaystyle \mathbf{y}}$

${\displaystyle F_{\mathbf{X}\mathbf{,}\mathbf{Y}}(\mathbf{x}\mathbf{,}\mathbf{y})=F_{\mathbf{X}}(\mathbf{x})\cdot F_{\mathbf{Y}}(\mathbf{y})}$

where${\displaystyle F_{\mathbf{X}}(\mathbf{x})}$ and${\displaystyle F_{\mathbf{Y}}(\mathbf{y})}$ denote the cumulative distribution functions of${\displaystyle \mathbf{X}}$ and${\displaystyle \mathbf{Y}}$ and${\displaystyle F_{\mathbf{X}\mathbf{,}\mathbf{Y}}(\mathbf{x}\mathbf{,}\mathbf{y})}$ denotes their joint cumulative distribution function. Independence of${\displaystyle \mathbf{X}}$ and${\displaystyle \mathbf{Y}}$ is often denoted by${\displaystyle \mathbf{X}\perp \perp \mathbf{Y}}$.Written component-wise,${\displaystyle \mathbf{X}}$ and${\displaystyle \mathbf{Y}}$ are called independent if for all${\displaystyle x_1,\dots ,x_m,y_1,\dots ,y_n}$

${\displaystyle F_{X_1,\dots ,X_m,Y_1,\dots ,Y_n}(x_1,\dots ,x_m,y_1,\dots ,y_n)=F_{X_1,\dots ,X_m}(x_1,\dots ,x_m)\cdot F_{Y_1,\dots ,Y_n}(y_1,\dots ,y_n)}$.

Thecharacteristic function of a random vector${\displaystyle \mathbf{X}}$ with${\displaystyle n}$ components is a function${\displaystyle {\mathbb{R}}^n\to \mathbb{C}}$ that maps every vector${\displaystyle \omega =({\omega }_1,\dots ,{\omega }_n{)}^T}$ to a complex number. It is defined by^{[2]: p. 468}

${\displaystyle {\phi }_{\mathbf{X}}(\omega )=\mathrm{E}\left[e^{i({\omega }^T\mathbf{X})}\right]=\mathrm{E}\left[e^{i({\omega }_1{X}_1+\dots +{\omega }_n{X}_n)}\right]}$.

One can take the expectation of aquadratic form in the random vector${\displaystyle \mathbf{X}}$ as follows:^{[5]: p.170–171}

${\displaystyle \mathrm{E}[{\mathbf{X}}^{T}A\mathbf{X}]=\mathrm{E}[\mathbf{X}{]}^{T}A\mathrm{E}[\mathbf{X}]+\mathrm{tr}(A{K}_{\mathbf{X}\mathbf{X}}),}$

where${\displaystyle K_{\mathbf{X}\mathbf{X}}}$ is the covariance matrix of${\displaystyle \mathbf{X}}$ and${\displaystyle \mathrm{tr}}$ refers to thetrace of a matrix — that is, to the sum of the elements on itsmain diagonal (from upper left to lower right). Since the quadratic form is a scalar, so is its expectation.

Proof: Let${\displaystyle \mathbf{z}}$ be an${\displaystyle m\times 1}$ random vector with${\displaystyle \mathrm{E}[\mathbf{z}]=\mu }$ and${\displaystyle \mathrm{Cov}[\mathbf{z}]=V}$ and let${\displaystyle A}$ be an${\displaystyle m\times m}$ non-stochastic matrix.

Then based on the formula for the covariance, if we denote${\displaystyle {\mathbf{z}}^T=\mathbf{X}}$ and${\displaystyle {\mathbf{z}}^T{A}^T=\mathbf{Y}}$, we see that:

${\displaystyle \mathrm{Cov}[\mathbf{X},\mathbf{Y}]=\mathrm{E}[\mathbf{X}{\mathbf{Y}}^T]-\mathrm{E}[\mathbf{X}]\mathrm{E}[\mathbf{Y}{]}^T}$

Hence

which leaves us to show that

${\displaystyle \mathrm{Cov}[z^T,z^T{A}^T]=\mathrm{tr}(AV).}$

This is true based on the fact that one cancyclically permute matrices when taking a trace without changing the end result (e.g.:${\displaystyle \mathrm{tr}(AB)=\mathrm{tr}(BA)}$).

We seethat

And since

${\displaystyle {\left(z-\mu \right)}^T\left(Az-A\mu \right)}$

is ascalar, then

${\displaystyle (z-\mu {)}^T(Az-A\mu )=\mathrm{tr}\left((z-\mu {)}^T(Az-A\mu )\right)=\mathrm{tr}\left((z-\mu {)}^{T}A(z-\mu )\right)}$

trivially. Using the permutation we get:

${\displaystyle \mathrm{tr}\left((z-\mu {)}^{T}A(z-\mu )\right)=\mathrm{tr}\left(A(z-\mu )(z-\mu {)}^T\right),}$

and by plugging this into the original formula we get:

One can take the expectation of the product of two different quadratic forms in a zero-meanGaussian random vector${\displaystyle \mathbf{X}}$ as follows:^{[5]: pp. 162–176}

${\displaystyle \mathrm{E}\left[({\mathbf{X}}^{T}A\mathbf{X})({\mathbf{X}}^{T}B\mathbf{X})\right]=2\mathrm{tr}(A{K}_{\mathbf{X}\mathbf{X}}B{K}_{\mathbf{X}\mathbf{X}})+\mathrm{tr}(A{K}_{\mathbf{X}\mathbf{X}})\mathrm{tr}(B{K}_{\mathbf{X}\mathbf{X}})}$

where again${\displaystyle K_{\mathbf{X}\mathbf{X}}}$ is the covariance matrix of${\displaystyle \mathbf{X}}$. Again, since both quadratic forms are scalars and hence their product is a scalar, the expectation of their product is also a scalar.

Inportfolio theory infinance, an objective often is to choose a portfolio of risky assets such that the distribution of the random portfolio return has desirable properties. For example, one might want to choose the portfolio return having the lowest variance for a given expected value. Here the random vector is the vector${\displaystyle \mathbf{r}}$ of random returns on the individual assets, and the portfolio returnp (a random scalar) is the inner product of the vector of random returns with a vectorw of portfolio weights — the fractions of the portfolio placed in the respective assets. Sincep =w^T${\displaystyle \mathbf{r}}$, the expected value of the portfolio return isw^TE(${\displaystyle \mathbf{r}}$) and the variance of the portfolio return can be shown to bew^TCw, where C is the covariance matrix of${\displaystyle \mathbf{r}}$.

Inlinear regression theory, we have data onn observations on a dependent variabley andn observations on each ofk independent variablesx_j. The observations on the dependent variable are stacked into a column vectory; the observations on each independent variable are also stacked into column vectors, and these latter column vectors are combined into adesign matrixX (not denoting a random vector in this context) of observations on the independent variables. Then the following regression equation is postulated as a description of the process that generated the data:

${\displaystyle y=X\beta +e,}$

where β is a postulated fixed but unknown vector ofk response coefficients, ande is an unknown random vector reflecting random influences on the dependent variable. By some chosen technique such asordinary least squares, a vector${\displaystyle \hat{\beta }}$ is chosen as an estimate of β, and the estimate of the vectore, denoted${\displaystyle \hat{e}}$, is computed as

${\displaystyle \hat{e}=y-X\hat{\beta }.}$

Then the statistician must analyze the properties of${\displaystyle \hat{\beta }}$ and${\displaystyle \hat{e}}$, which are viewed as random vectors since a randomly different selection ofn cases to observe would have resulted in different values for them.

The evolution of ak×1 random vector${\displaystyle \mathbf{X}}$ through time can be modelled as avector autoregression (VAR) as follows:

${\displaystyle {\mathbf{X}}_t=c+A_1{\mathbf{X}}_{t-1}+A_2{\mathbf{X}}_{t-2}+\cdots +A_p{\mathbf{X}}_{t-p}+{\mathbf{e}}_t,}$

where thei-periods-back vector observation${\displaystyle {\mathbf{X}}_{t-i}}$ is called thei-th lag of${\displaystyle \mathbf{X}}$,c is ak × 1 vector of constants (intercepts),A_i is a time-invariantk × kmatrix and${\displaystyle {\mathbf{e}}_t}$ is ak × 1 random vector oferror terms.

• Stark, Henry; Woods, John W. (2012). "Random Vectors".Probability, Statistics, and Random Processes for Engineers (Fourth ed.). Pearson. pp. 295–339.ISBN 978-0-13-231123-6.

• Multivariate statistics
• Algebra of random variables

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