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Joint probability mass function

The joint probability mass function is a function that completely characterizes the distribution of a discrete random vector. When evaluated at a given point, it gives the probability that the realization of the random vector will be equal to that point.

Synonyms and acronyms

The term joint probability function is often used as a synonym. Sometimes, the abbreviation joint pmf is used.


The following is a formal definition.

Definition Let X be a Kx1 discrete random vector. Its joint probability mass function is a function [eq1] such that[eq2]where [eq3] is the probability that the random vector X takes the value x.


Suppose X is a $2\times 1$ discrete random vector and that its support (the set of values it can take) is:[eq4]If the three values have the same probability, then the joint probability mass function is:[eq5]Denoting the two components of X by X_1 and X_2, its joint pmf can also be written using the following alternative notation:[eq6]

More details

This is a glossary entry. For a thorough discussion of joint pmfs, go to the lecture entitled Random vectors, where discrete random vectors are introduced and you can also find some exercises involving joint pmfs.

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