Multivariate random variable


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 whose components are scalar-valued random variables on the same probability space as each other,, where is the sample space, is the sigma-algebra, and is the probability measure.

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 :
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 of an random vector is an matrix whose 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 is the matrix
where again the matrix expectation is taken element-by-element in the matrix. Here the 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 of an random vector is an matrix whose 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 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. 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.
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 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 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 as follows:
where the i-periods-back vector observation is called the i-th lag of, c is a k × 1 vector of constants, Ai is a time-invariant k × k matrix and is a k × 1 random vector of error terms.