In probability and statistics, a multivariate random variable or random vector is a list or vector of mathematical variabless 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.

Formally, a multivariate random variable is a column vector (or its transpose, which is a row vector) whose components are random variables on the probability space is the sample space is the sigma-algebra is the probability measure (a function returning each event's probability).

Probability distribution

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 are called marginal distributions. The conditional probability distribution is known to be a particular value.

The cumulative distribution function is defined as

.

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

can be defined by applying an affine transformation:

1. column vector.

is an invertible matrix is

1. .

Invertible mappings

More generally we can study invertible mappings of random vectors.

and let the Jacobian determinant is of probability density

denotes the indicator function.

Expected value

The expected value whose elements are the expected values of the respective random variables.

Covariance and cross-covariance

Definitions

The covariance matrix (also called second central moment matrix whose (i,j)th element is the covariance between the i th and the j th matrix computed as, where the superscript T refers to the transpose of the indicated vector:

By extension, the cross-covariance matrix 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 and the j th.

Properties

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

1. .

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

1. .

, i.e.

1. .

Uncorrelatedness

are called uncorrelated if

1. .

They are uncorrelated if and only if is zero.

Correlation and cross-correlation

Definitions

The correlation matrix (also called second moment matrix whose (i,j)th element is the correlation between the i th and the j th, where the superscript T refers to the transpose of the indicated vector:

By extension, the cross-correlation matrix matrix

Properties

The correlation matrix is related to the covariance matrix by

1. .

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

Orthogonality

are called orthogonal if

1. .

Independence

are called independent

denotes their joint cumulative distribution function

1. .

Characteristic function

The characteristic function to a complex number. It is defined by

1. .

Further properties

Expectation of a quadratic form

One can take the expectation of a quadratic form as follows:

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 non-stochastic matrix.

, 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).

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 as follows:

. 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 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 wT) and the variance of the portfolio return can be shown to be wTCw.

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

, 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 through time can be modelled as a vector autoregression (VAR) as follows:

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

References

Further reading

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