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Joint probability mass function

by , PhD

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.

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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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