StatlectThe Digital Textbook

# Discrete random vector

A discrete random vector is a random vector that can take either a finite or an infinite but countable number of values.

## Definition

The following is a possible definition.

Definition A random vector is said to be discrete if and only if the set of values it can take has either a finite or an infinite but countable number of elements, and there exists a function , called joint probability mass function, such thatwhere is the probability that takes the value .

A random vector that is discrete is also said to possess a multivariate discrete distribution.

## Example

Suppose a random vector can take only one of three values, , or defined byand that has probability of being observed, while and have probability . Then, is discrete because it can take only finitely many values. Its joint probability mass function isWhen the two entries of are denoted by and , the joint probability mass function can also be written as

## More details

The lecture entitled Random vectors provides a more complete treatment of discrete random vectors and joint probability mass functions.