##
Overview
Section* *

As the title of the lesson suggests, in this lesson, we'll learn how to extend the concept of a probability distribution of one random variable \(X\) to a joint probability distribution of two random variables \(X\) and \(Y\). In some cases, \(X\) and \(Y\)may both be discrete random variables. For example, suppose \(X\) denotes the number of significant others a randomly selected person has, and \(Y\) denotes the number of arguments the person has each week. We might want to know if there is a relationship between \(X\) and \(Y\). Or, we might want to know the probability that \(X\) takes on a particular value \(x\) and \(Y\) takes on a particular value \(y\). That is, we might want to know \(P(X=x, Y=y)\).

## Objectives

- To learn the formal definition of a joint probability mass function of two discrete random variables.
- To learn how to use a joint probability mass function to find the probability of a specific event.
- To learn how to find a marginal probability mass function of a discrete random variable \(X\) from the joint probability mass function of \(X\) and \(Y\).
- To learn a formal definition of the independence of two random variables \(X\) and \(Y\).
- To learn how to find the expectation of a function of the discrete random variables \(X\) and \(Y\) using their joint probability mass function.
- To learn how to find the means and variances of the discrete random variables \(X\) and \(Y\) using their joint probability mass function.
- To learn what it means that \(X\) and \(Y\) have a joint triangular support.
- To learn that, in general, any two random variables \(X\) and \(Y\) having a joint triangular support must be dependent.
- To learn what it means that \(X\) and \(Y\) have a joint rectangular support.
- To learn that, in general, any two random variables \(X\) and \(Y\) having a joint rectangular support may or may not be independent.
- To learn about the trinomial distribution.
- To be able to apply the methods learned in the lesson to new problems.