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Joint distribution function

The joint distribution function completely characterizes the probability distribution of a random vector X. When evaluated$\,$ at the point x, it gives the probability that each component of X takes on a value smaller than or equal to the respective component of x.

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Synonyms and acronyms

It is also called joint cumulative distribution function (abbreviated as joint cdf).

Defintion

The following is a formal definition.

Definition The joint distribution function of a Kx1 random vector X is a function [eq1] such that:[eq2]where the components of X and x are denoted by $X_{k}$ and $x_{k}$ respectively, for $k=1,\ldots ,K$.

Usage

The joint distribution function can be used to the derive the marginal distributions of the single components of the random vector (see Random vectors). It is also used to check whether two ore more random variables are independent (see Independent random variables).

More details

More details about joint distribution functions can be found in the lecture entitled Random vectors.

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