# Lesson 18: The Correlation Coefficient

### Introduction

In the previous lesson, we learned about the joint probability distribution of two random variables *X* and *Y. *In this lesson, we'll extend our investigation of the relationship between two random variables by learning how to quantify the *extent* or *degree* to which two random variables *X* and *Y* are associated or **correlated**. For example, Suppose *X* denotes the number of cups of hot chocolate sold daily at a local café, and *Y* denotes the number of apple cinnamon muffins sold daily at the same café. Then, the manager of the café might benefit from knowing whether *X* and *Y* are highly correlated or not. If the random variables are highly correlated, then the manager would know to make sure that both are available on a given day. If the random variables are not highly correlated, then the manager would know that it would be okay to have one of the items available without the other. As the title of the lesson suggests, the **correlation coefficient** is the statistical measure that is going to allow us to quantify the degree of correlation between two random variables *X* and *Y*.

### Objectives

- To learn a formal definition of the covariance between two random variables
*X*and*Y*. - To learn how to calculate the covariance between any two random variables
*X*and*Y*. - To learn a shortcut, or alternative, formula for the covariance between two random variables
*X*and*Y*. - To learn a formal definition of the correlation coefficient between two random variables
*X*and*Y*. - To learn how to calculate the correlation coefficient between any two random variables
*X*and*Y*. - To learn how to interpret the correlation coefficient between any two random variables
*X*and*Y*. - To learn that if
*X*and*Y*are independent random variables, then the covariance and correlation between*X*and*Y*are both zero. - To learn that if the correlation between
*X*and*Y*is 0, then*X*and*Y*are not necessarily independent. - To learn how the correlation coefficient gets its sign.
- To learn that the correlation coefficient measures the strength of the
*linear*relationship between two random variables*X*and*Y*. - To learn that the correlation coefficient is necessarily a number between −1 and +1.
- To understand the steps involved in each of the proofs in the lesson.
- To be able to apply the methods learned in the lesson to new problems.