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Continuous random vector

by , PhD

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

A random vector has the following characteristics:

Table of Contents


A continuous random vector is sometimes also called absolutely continuous.


The following is a formal definition.

Definition A Kx1 random vector X is said to be 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.


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

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Previous entry: Absolutely continuous random variable

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How to cite

Please cite as:

Taboga, Marco (2021). "Continuous random vector", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix.

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