# Lesson 17: Distributions of Two Discrete Random Variables

### Introduction

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.