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.

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

The following is a formal definition.

Definition A random vector is said to be absolutely 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.

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:

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

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

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