Open main menu

Multivariate random variable

  (Redirected from Random vector)

In probability, and statistics, a multivariate random variable or random vector is a list 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 random variables, e.g. a random matrix, random tree, random sequence, stochastic process, etc.

More formally, a multivariate random variable is a column vector (or its transpose, which is a row vector) whose components are scalar-valued random variables on the same probability space as each other, , where is the sample space, is the sigma-algebra (the collection of all events), and is the probability measure (a function returning each event's probability).

Probability distributionEdit

Every random vector gives rise to a probability measure on   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   are called marginal distributions. The conditional probability distribution of   given   is the probability distribution of   when   is known to be a particular value.

The cumulative distribution function   of a random vector   is defined as[1]:p.15

 

 

 

 

 

(Eq.1)

where  .

Operations on random vectorsEdit

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 inner products.

Affine transformationsEdit

Similarly, a new random vector   can be defined by applying an affine transformation   to a random vector  :

 , where   is an   matrix and   is an   column vector.

If   is an invertible matrix and   has a probability density function  , then the probability density of   is

 .

Invertible mappingsEdit

More generally we can study invertible mappings of random vectors.[2]:p.290–291

Let   be a one-to-one mapping from an open subset   of   onto a subset   of  , let   have continuous partial derivatives in   and let the Jacobian determinant of   be zero at no point of  . Assume that the real random vector   has a probability density function   and satisfies  . Then the random vector   is of probability density

 

where   denotes the indicator function and set   denotes support of  .

Expected valueEdit

The expected value or mean of a random vector   is a fixed vector   whose elements are the expected values of the respective random variables.[3]:p.333

 

 

 

 

 

(Eq.2)

Covariance and cross-covarianceEdit

DefinitionsEdit

The covariance matrix (also called second central moment or variance-covariance matrix) of an   random vector is an   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   matrix computed as  , where the superscript T refers to the transpose of the indicated vector:[2]:p. 464[3]:p.335

 

 

 

 

 

(Eq.3)

By extension, the cross-covariance matrix between two random vectors   and   (  having   elements and   having   elements) is the   matrix[3]:p.336

 

 

 

 

 

(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 the i th element of   and the j th element of  .

PropertiesEdit

The covariance matrix is a symmetric matrix, i.e.[2]:p. 466

 .

The covariance matrix is a positive semidefinite matrix, i.e.[2]:p. 465

 .

The cross-covariance matrix   is simply the transpose of the matrix  , i.e.

 .

UncorrelatednessEdit

Two random vectors   and   are called uncorrelated if

 .

They are uncorrelated if and only if their cross-covariance matrix   is zero.[3]:p.337

Correlation and cross-correlationEdit

DefinitionsEdit

The correlation matrix (also called second moment) of an   random vector is an   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   matrix computed as  , where the superscript T refers to the transpose of the indicated vector[4]:p.190[3]:p.334:

 

 

 

 

 

(Eq.5)

By extension, the cross-correlation matrix between two random vectors   and   (  having   elements and   having   elements) is the   matrix

 

 

 

 

 

(Eq.6)

PropertiesEdit

The correlation matrix is related to the covariance matrix by

 .

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

 

OrthogonalityEdit

Two random vectors of the same size   and   are called orthogonal if

 .

IndependenceEdit

Two random vectors   and   are called independent if for all   and  

 

where   and   denote the cumulative distribution functions of   and   and  denotes their joint cumulative distribution function. Independence of   and   is often denoted by  . Written component-wise,   and   are called independent if for all  

 .

Characteristic functionEdit

The characteristic function of a random vector   with   components is a function   that maps every vector   to a complex rumber. It is defined by[2]:p. 468

 .

Further propertiesEdit

Expectation of a quadratic formEdit

One can take the expectation of a quadratic form in the random vector   as follows:[5]:p.170–171

 

where   is the covariance matrix of   and   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   be an   random vector with   and   and let   be an   non-stochastic matrix.

Then based on the formula for the covariance, if we denote   and  , we see that:

 

Hence

 

which leaves us to show that

 

This is true based on the fact that one can cyclically permute matrices when taking a trace without changing the end result (e.g.:  ).

We see that

 

And since

 

is a scalar, then

 

trivially. Using the permutation we get:

 

and by plugging this into the original formula we get:

 

Expectation of the product of two different quadratic formsEdit

One can take the expectation of the product of two different quadratic forms in a zero-mean Gaussian random vector   as follows:[5]:pp. 162–176

 

where again   is the covariance matrix of  . Again, since both quadratic forms are scalars and hence their product is a scalar, the expectation of their product is also a scalar.

ApplicationsEdit

Portfolio theoryEdit

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   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 = wT , the expected value of the portfolio return is wTE( ) and the variance of the portfolio return can be shown to be wTCw, where C is the covariance matrix of  .

Regression theoryEdit

In linear regression theory, we have data on n observations on a dependent variable y and n observations on each of k independent variables xj. 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:

 

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   is chosen as an estimate of β, and the estimate of the vector e, denoted  , is computed as

 

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

Vector time seriesEdit

The evolution of a k×1 random vector   through time can be modelled as a vector autoregression (VAR) as follows:

 

where the i-periods-back vector observation   is called the i-th lag of  , c is a k × 1 vector of constants (intercepts), Ai is a time-invariant k × k matrix and   is a k × 1 random vector of error terms.

ReferencesEdit

  1. ^ Gallager, Robert G. (2013). Stochastic Porcesses Theory for Applications. Cambridge University Press. ISBN 978-1-107-03975-9.
  2. ^ a b c d e Lapidoth, Amos, A foundation in Digital Communication, Cambridge University Press, 2009.
  3. ^ a b c d e Gubner, John A. (2006). Probability and Random Processes for Electrical and Computer Engineers. Cambridge University Press. ISBN 978-0-521-86470-1.
  4. ^ Papoulis, Athanasius, Probability, Random variables and Stochastic processes, McGraw-Hill, 1991
  5. ^ a b Kendrick, David, Stochastic Control for Economic Models, McGraw-Hill, 1981.