Multivariate random variable

 ( Multivariate Statistics ) 

In probability, and statistics, a multivariate random variable or random vector is a list or vector of mathematical variables 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 individual statistical unit. For example, while a given person has a specific age, height and weight, the representation of these features of an unspecified person from within a group would be a random vector. Normally each element of a random vector is a real number.

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

Formally, a multivariate random variable is a column vector $\backslash mathbf\{X\}\; =\; (X\_1,\backslash dots,X\_n)^\backslash mathsf\{T\}$ (or its transpose, which is a row vector) whose components are on the probability space $(\backslash Omega,\; \backslash mathcal\{F\},\; P)$, where $\backslash Omega$ is the sample space, $\backslash mathcal\{F\}$ is the sigma-algebra (the collection of all events), and $P$ is the probability measure (a function returning each event's probability).

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

The distributions of each of the component random variables $X\_i$ are called marginal distributions. The conditional probability distribution of $X\_i$ given $X\_j$ is the probability distribution of $X\_i$ when $X\_j$ is known to be a particular value.

The cumulative distribution function $F\_\{\backslash mathbf\{X\}\}\; :\; \backslash R^n\; \backslash mapsto\; 0,1$ of a random vector $\backslash mathbf\{X\}=(X\_1,\backslash dots,X\_n)^\backslash mathsf\{T\}$ is defined as

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where $\backslash mathbf\{x\}\; =\; (x\_1,\; \backslash dots,\; x\_n)^\backslash mathsf\{T\}$.

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Operations on random vectors Random vectors can be subjected to the same kinds of algebraic operations as can non-random vectors: addition, subtraction, multiplication by a scalar, and the taking of Dot product.

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Affine transformations Similarly, a new random vector $\backslash mathbf\{Y\}$ can be defined by applying an affine transformation $g\backslash colon\; \backslash mathbb\{R\}^n\; \backslash to\; \backslash mathbb\{R\}^n$ to a random vector $\backslash mathbf\{X\}$:

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

If $\backslash mathbf\{A\}$ is an invertible matrix and $\backslash textstyle\backslash mathbf\{X\}$ has a probability density function $f\_\{\backslash mathbf\{X\}\}$, then the probability density of $\backslash mathbf\{Y\}$ is

$f\_\{\backslash mathbf\{Y\}\}(y)=\backslash frac\{f\_\{\backslash mathbf\{X\}\}(\backslash mathbf\{A\}^\{-1\}(y-b))\}$.

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Invertible mappings More generally we can study invertible mappings of random vectors.

Marco Taboga (2025). 9781981369195, CreateSpace Independent Publishing Platform. ISBN 9781981369195

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

$\backslash left.\; f\_\{\backslash mathbf\{Y\}\}(\backslash mathbf\{y\})=\backslash frac\{f\_\{\backslash mathbf\{X\}\}(\backslash mathbf\{x\})\}\{\backslash left\; |\backslash det\backslash left\; (\backslash frac\{\backslash partial\; \backslash mathbf\{y\}\}\{\backslash partial\; \backslash mathbf\{x\}\}\backslash right\; )\backslash right\; |\}\; \backslash right\; |\_\{\backslash mathbf\{x\}=g^\{-1\}(\backslash mathbf\{y\})\}\; \backslash mathbf\{1\}(\backslash mathbf\{y\}\; \backslash in\; R\_\backslash mathbf\{Y\})$

where $\backslash mathbf\{1\}$ denotes the indicator function and set $R\_\backslash mathbf\{Y\}\; =\; \backslash \{\; \backslash mathbf\{y\}\; =\; g(\backslash mathbf\{x\}):\; f\_\{\backslash mathbf\{X\}\}(\backslash mathbf\{x\})\; >\; 0\; \backslash \}\; \backslash subseteq\; \backslash mathcal\{R\}$ denotes support of $\backslash mathbf\{Y\}$.

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Expected value The expected value or mean of a random vector $\backslash mathbf\{X\}$ is a fixed vector $\backslash operatorname\{E\}\backslash mathbf\{X\}$ whose elements are the expected values of the respective random variables.

John A. Gubner (2025). 9780521864701, Cambridge University Press. ISBN 9780521864701

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Covariance and cross-covariance

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Definitions The covariance matrix (also called second central moment or variance-covariance matrix) of an $n\; \backslash times\; 1$ random vector is an $n\; \backslash times\; n$ matrix whose ( i,j)^th element is the covariance between the i ^th and the j ^th random variables. The covariance matrix is the expected value, element by element, of the $n\; \backslash times\; n$ matrix computed as $\backslash mathbf\{X\}-\backslash operatorname\{E\}[\backslash mathbf\{X\}]\; \backslash mathbf\{X\}-\backslash operatorname\{E\}[\backslash mathbf\{X\}]^T$, where the superscript T refers to the transpose of the indicated vector:

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By extension, the cross-covariance matrix between two random vectors $\backslash mathbf\{X\}$ and $\backslash mathbf\{Y\}$ ($\backslash mathbf\{X\}$ having $n$ elements and $\backslash mathbf\{Y\}$ having $p$ elements) is the $n\; \backslash times\; p$ matrix

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

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Properties

The covariance matrix is a symmetric matrix, i.e.

$\backslash operatorname\{K\}\_\{\backslash mathbf\{X\}\backslash mathbf\{X\}\}^T\; =\; \backslash operatorname\{K\}\_\{\backslash mathbf\{X\}\backslash mathbf\{X\}\}$.

The covariance matrix is a positive semidefinite matrix, i.e.

$\backslash mathbf\{a\}^T\; \backslash operatorname\{K\}\_\{\backslash mathbf\{X\}\backslash mathbf\{X\}\}\; \backslash mathbf\{a\}\; \backslash ge\; 0\; \backslash quad\; \backslash text\{for\; all\; \}\; \backslash mathbf\{a\}\; \backslash in\; \backslash mathbb\{R\}^n$.

The cross-covariance matrix $\backslash operatorname\{Cov\}\backslash mathbf\{Y\},\backslash mathbf\{X\}$ is simply the transpose of the matrix $\backslash operatorname\{Cov\}\backslash mathbf\{X\},\backslash mathbf\{Y\}$, i.e.

$\backslash operatorname\{K\}\_\{\backslash mathbf\{Y\}\backslash mathbf\{X\}\}\; =\; \backslash operatorname\{K\}\_\{\backslash mathbf\{X\}\backslash mathbf\{Y\}\}^T$.

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Uncorrelatedness

Two random vectors $\backslash mathbf\{X\}=(X\_1,...,X\_m)^T$ and $\backslash mathbf\{Y\}=(Y\_1,...,Y\_n)^T$ are called uncorrelated if

$\backslash operatorname\{E\}\backslash mathbf\{X\}\; =\; \backslash operatorname\{E\}\backslash mathbf\{X\}\backslash operatorname\{E\}\backslash mathbf\{Y\}^T$.

They are uncorrelated if and only if their cross-covariance matrix $\backslash operatorname\{K\}\_\{\backslash mathbf\{X\}\backslash mathbf\{Y\}\}$ is zero.

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Correlation and cross-correlation

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Definitions

The correlation matrix (also called second moment) of an $n\; \backslash times\; 1$ random vector is an $n\; \backslash times\; n$ matrix whose ( i,j)^th element is the correlation between the i ^th and the j ^th random variables. The correlation matrix is the expected value, element by element, of the $n\; \backslash times\; n$ matrix computed as $\backslash mathbf\{X\}\; \backslash mathbf\{X\}^T$, where the superscript T refers to the transpose of the indicated vector:

Athanasius Papoulis (1991). 9780070484771, McGraw-Hill. ISBN 9780070484771

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

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Properties

The correlation matrix is related to the covariance matrix by

$\backslash operatorname\{R\}\_\{\backslash mathbf\{X\}\backslash mathbf\{X\}\}\; =\; \backslash operatorname\{K\}\_\{\backslash mathbf\{X\}\backslash mathbf\{X\}\}\; +\; \backslash operatorname\{E\}\backslash mathbf\{X\}\backslash operatorname\{E\}\backslash mathbf\{X\}^T$.

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

$\backslash operatorname\{R\}\_\{\backslash mathbf\{X\}\backslash mathbf\{Y\}\}\; =\; \backslash operatorname\{K\}\_\{\backslash mathbf\{X\}\backslash mathbf\{Y\}\}\; +\; \backslash operatorname\{E\}\backslash mathbf\{X\}\backslash operatorname\{E\}\backslash mathbf\{Y\}^T$

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Orthogonality

Two random vectors of the same size $\backslash mathbf\{X\}=(X\_1,...,X\_n)^T$ and $\backslash mathbf\{Y\}=(Y\_1,...,Y\_n)^T$ are called orthogonal if

$\backslash operatorname\{E\}\backslash mathbf\{X\}^T\; =\; 0$.

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Independence

Two random vectors $\backslash mathbf\{X\}$ and $\backslash mathbf\{Y\}$ are called independent if for all $\backslash mathbf\{x\}$ and $\backslash mathbf\{y\}$

$F\_\{\backslash mathbf\{X,Y\}\}(\backslash mathbf\{x,y\})\; =\; F\_\{\backslash mathbf\{X\}\}(\backslash mathbf\{x\})\; \backslash cdot\; F\_\{\backslash mathbf\{Y\}\}(\backslash mathbf\{y\})$

where $F\_\{\backslash mathbf\{X\}\}(\backslash mathbf\{x\})$ and $F\_\{\backslash mathbf\{Y\}\}(\backslash mathbf\{y\})$ denote the cumulative distribution functions of $\backslash mathbf\{X\}$ and $\backslash mathbf\{Y\}$ and$F\_\{\backslash mathbf\{X,Y\}\}(\backslash mathbf\{x,y\})$ denotes their joint cumulative distribution function. Independence of $\backslash mathbf\{X\}$ and $\backslash mathbf\{Y\}$ is often denoted by $\backslash mathbf\{X\}\; \backslash perp\backslash !\backslash !\backslash !\backslash perp\; \backslash mathbf\{Y\}$. Written component-wise, $\backslash mathbf\{X\}$ and $\backslash mathbf\{Y\}$ are called independent if for all $x\_1,\backslash ldots,x\_m,y\_1,\backslash ldots,y\_n$

$F\_\{X\_1,\backslash ldots,X\_m,Y\_1,\backslash ldots,Y\_n\}(x\_1,\backslash ldots,x\_m,y\_1,\backslash ldots,y\_n)\; =\; F\_\{X\_1,\backslash ldots,X\_m\}(x\_1,\backslash ldots,x\_m)\; \backslash cdot\; F\_\{Y\_1,\backslash ldots,Y\_n\}(y\_1,\backslash ldots,y\_n)$.

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Characteristic function

The characteristic function of a random vector $\backslash mathbf\{X\}$ with $n$ components is a function $\backslash mathbb\{R\}^n\; \backslash to\; \backslash mathbb\{C\}$ that maps every vector $\backslash mathbf\{\backslash omega\}\; =\; (\backslash omega\_1,\backslash ldots,\backslash omega\_n)^T$ to a complex number. It is defined by

$\backslash varphi\_\{\backslash mathbf\{X\}\}(\backslash mathbf\{\backslash omega\})\; =\; \backslash operatorname\{E\}\; \backslash left\; =\; \backslash operatorname\{E\}\; \backslash left$.

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Further properties

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Expectation of a quadratic form

One can take the expectation of a quadratic form in the random vector $\backslash mathbf\{X\}$ as follows:

David Kendrick (1981). 9780070339620, McGraw-Hill. ISBN 9780070339620

$\backslash operatorname\{E\}\backslash mathbf\{X\}^\{T\}A\backslash mathbf\{X\}\; =\; \backslash operatorname\{E\}\backslash mathbf\{X\}^\{T\}A\backslash operatorname\{E\}\backslash mathbf\{X\}\; +\; \backslash operatorname\{tr\}(A\; K\_\{\backslash mathbf\{X\}\backslash mathbf\{X\}\}),$

where $K\_\{\backslash mathbf\{X\}\backslash mathbf\{X\}\}$ is the covariance matrix of $\backslash mathbf\{X\}$ and $\backslash operatorname\{tr\}$ refers to the trace of a matrix — that is, to the sum of the elements on its main diagonal (from upper left to lower right). Since the quadratic form is a scalar, so is its expectation.

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

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

$\backslash operatorname\{Cov\}\backslash mathbf\{X\},\backslash mathbf\{Y\}\; =\; \backslash operatorname\{E\}\backslash mathbf\{X\}\backslash mathbf\{Y\}^T-\backslash operatorname\{E\}\backslash mathbf\{X\}\backslash operatorname\{E\}\backslash mathbf\{Y\}^T$

Hence

$\backslash begin\{align\}$

\operatorname{E}XY^T &= \operatorname{Cov}X,Y+\operatorname{E}X\operatorname{E}Y^T \\ \operatorname{E}z^T &= \operatorname{Cov}z^T,z^T + \operatorname{E}z^T\operatorname{E}z^T^T \\ &=\operatorname{Cov}z^T + \mu^T (\mu^T A^T)^T \\ &=\operatorname{Cov}z^T + \mu^T A \mu , \end{align}

which leaves us to show that

$\backslash operatorname\{Cov\}z^T=\backslash operatorname\{tr\}(AV).$

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

We see that

$\backslash begin\{align\}$

\operatorname{Cov}z^T,z^T &= \operatorname{E} \left\left(z^T \\ &= \operatorname{E} \left\\ &= \operatorname{E} \left. \end{align}

And since

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

is a scalar, then

$(z\; -\; \backslash mu)^T\; (\; Az\; -\; A\backslash mu)=\; \backslash operatorname\{tr\}\backslash left(\; \{(z\; -\; \backslash mu\; )^T\; (Az\; -\; A\backslash mu\; )\}\; \backslash right)\; =\; \backslash operatorname\{tr\}\; \backslash left((z\; -\; \backslash mu\; )^T\; A(z\; -\; \backslash mu\; )\; \backslash right)$

trivially. Using the permutation we get:

$\backslash operatorname\{tr\}\backslash left(\; \{(z\; -\; \backslash mu\; )^T\; A(z\; -\; \backslash mu\; )\}\; \backslash right)\; =\; \backslash operatorname\{tr\}\backslash left(\; \{A(z\; -\; \backslash mu\; )(z\; -\; \backslash mu\; )^T\}\; \backslash right),$

and by plugging this into the original formula we get:

$\backslash begin\{align\}$

\operatorname{Cov} \left &= E\left \\ &= E \left \\ &= \operatorname{tr} \left( {A \cdot \operatorname{E} \left((z - \mu )(z - \mu )^T \right) } \right) \\ &= \operatorname{tr} (A V). \end{align}

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Expectation of the product of two different quadratic forms

One can take the expectation of the product of two different quadratic forms in a zero-mean Joint normality random vector $\backslash mathbf\{X\}$ as follows:

$\backslash operatorname\{E\}\backslash left(\backslash mathbf\{X\}^\{T\}A\backslash mathbf\{X\})(\backslash mathbf\{X\}^\{T\}B\backslash mathbf\{X\})\backslash right\; =\; 2\backslash operatorname\{tr\}(A\; K\_\{\backslash mathbf\{X\}\backslash mathbf\{X\}\}\; B\; K\_\{\backslash mathbf\{X\}\backslash mathbf\{X\}\})\; +\; \backslash operatorname\{tr\}(A\; K\_\{\backslash mathbf\{X\}\backslash mathbf\{X\}\})\backslash operatorname\{tr\}(B\; K\_\{\backslash mathbf\{X\}\backslash mathbf\{X\}\})$

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

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Applications

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Portfolio theory

In portfolio theory in finance, 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 $\backslash mathbf\{r\}$ of random returns on the individual assets, and the portfolio return p (a random scalar) is the inner product of the vector of random returns with a vector w of portfolio weights — the fractions of the portfolio placed in the respective assets. Since p = w^T$\backslash mathbf\{r\}$, the expected value of the portfolio return is w^TE($\backslash mathbf\{r\}$) and the variance of the portfolio return can be shown to be w^TC w, where C is the covariance matrix of $\backslash mathbf\{r\}$.

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Regression theory

In linear regression theory, we have data on n observations on a dependent variable y and n observations on each of k independent variables x_j. The observations on the dependent variable are stacked into a column vector y; the observations on each independent variable are also stacked into column vectors, and these latter column vectors are combined into a design matrix X (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:

$y\; =\; X\; \backslash beta\; +\; e,$

where β is a postulated fixed but unknown vector of k response coefficients, and e is an unknown random vector reflecting random influences on the dependent variable. By some chosen technique such as ordinary least squares, a vector $\backslash hat\; \backslash beta$ is chosen as an estimate of β, and the estimate of the vector e, denoted $\backslash hat\; e$, is computed as

$\backslash hat\; e\; =\; y\; -\; X\; \backslash hat\; \backslash beta.$

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

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Vector time series

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

$\backslash mathbf\{X\}\_t\; =\; c\; +\; A\_1\; \backslash mathbf\{X\}\_\{t-1\}\; +\; A\_2\; \backslash mathbf\{X\}\_\{t-2\}\; +\; \backslash cdots\; +\; A\_p\; \backslash mathbf\{X\}\_\{t-p\}\; +\; \backslash mathbf\{e\}\_t,\; \backslash ,$

where the i-periods-back vector observation $\backslash mathbf\{X\}\_\{t-i\}$ is called the i-th lag of $\backslash mathbf\{X\}$, c is a k × 1 vector of constants (Y-intercept), A_i is a time-invariant k ×  k matrix and $\backslash mathbf\{e\}\_t$ is a k × 1 random vector of error terms.

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Further reading

• Henry Stark (2025). 9780132311236, Pearson. ISBN 9780132311236

Categories: Multivariate Statistics, Algebra Of Random Variables

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