# 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.

## Definition

The following is a formal definition.

Definition Let be a discrete random vector. Its joint probability mass function is a function such thatwhere is the probability that the random vector takes the value .

## Example

Suppose is a discrete random vector and that its support (the set of values it can take) is:If the three values have the same probability, then the joint probability mass function is:Denoting the two components of by and , its joint pmf can also be written using the following alternative notation:

## 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.