Multivariate random variable
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 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 distribution

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

where . 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 inner products. Affine transformations
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 mappings
More generally we can study invertible mappings of random vectors. 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 value
The expected value or mean of a random vector  is a fixed vector  whose elements are the expected values of the respective random variables. Covariance and cross-covariance

Definitions
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:

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

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 . Properties
The covariance matrix is a symmetric matrix, i.e. . The covariance matrix is a positive semidefinite matrix, i.e. . The cross-covariance matrix  is simply the transpose of the matrix , i.e. . Uncorrelatedness
Two random vectors  and  are called uncorrelated if
. They are uncorrelated if and only if their cross-covariance matrix  is zero. Correlation and cross-correlation

Definitions
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:

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

Properties
The correlation matrix is related to the covariance matrix by
. Similarly for the cross-correlation matrix and the cross-covariance matrix:

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

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 function
The characteristic function of a random vector  with  components is a function  that maps every vector  to a complex number. It is defined by

. Further properties

Expectation of a quadratic form
One can take the expectation of a quadratic form in the random vector  as follows:

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 forms
One can take the expectation of the product of two different quadratic forms in a zero-mean Gaussian random vector  as follows:

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. Applications

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  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 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 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 series

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. References

Further reading

Multivariate statistics
Algebra of random variables

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