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Absolutely continuous random vector

The concept of absolutely continuous random vector is a multivariate generalization of the concept of absolutely continuous random variable.

A random vector is absolutely continuous if the set of values it can take is not countable and the probability that its realization will belong to a given set can be computed as a multiple integral over that set of a function called joint probability density function.

Synonyms

An absolutely continuous random vector is also simply called continuous random vector, omitting the adverb "absolutely".

Definition

The following is a formal definition.

Definition A Kx1 random vector X is said to be absolutely continuous if the set of values it can take is not countable and the probability that X takes a value in a given hyper-rectangle[eq1]can be expressed as a multiple integral:[eq2]where the integrand function [eq3] is called the joint probability density function of X.

With vector notation, the multiple integral above could also be written in more compact form as[eq4]which makes clear that the joint probability density function is a straightforward multivariate generalization of the univariate probability density function.

A continuous random vector is often said to have a multivariate continuous distribution.

Example

Let X be a $2	imes 1$ continuous random vector that can take values in the set [eq5]Let its joint probability density function be[eq6]Suppose we need to compute the probability[eq7]This can be calculated as a multiple integral:[eq8]

Common multivariate continuous distributions

The multivariate normal distribution and the multivariate Student distribution are examples of multivariate continuous distributions.

More details

More details about absolutely continuous random vectors can be found in the lecture entitled Random vectors.

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