 StatLect

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-rectangle can 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 as 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 be a continuous random vector that can take values in the set Let its joint probability density function be Suppose we need to compute the probability This 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.