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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.

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Definition

The following is a possible definition.

Definition A Kx1 random vector X 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 [eq1], called joint probability mass function, such that[eq2]where [eq3] is the probability that X takes the value x.

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

Example

Suppose a $1\times 2$ random vector X can take only one of three values, $a $, $b$ or $c$ defined by[eq4]and that a has probability $1/2$ of being observed, while $b$ and $c$ have probability $1/4$. Then, X is discrete because it can take only finitely many values. Its joint probability mass function is[eq5]When the two entries of x are denoted by $x_{1}$ and $x_{2}$, the joint probability mass function can also be written as[eq6]

More details

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

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