# Continuous random vector

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

A random vector has the following characteristics:

• the set of values it can take is not countable;

• 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

A continuous random vector is sometimes also called absolutely continuous.

## Definition

The following is a formal definition.

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

With vector notation, the multiple integral above could also be written in more compact form aswhich 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 be a continuous random vector that can take values in the set Let its joint probability density function beSuppose we need to compute the probabilityThis can be calculated as a multiple integral:

## Common multivariate continuous distributions

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

## More details

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