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