# Lesson 20: Distributions of Two Continuous Random Variables

### Introduction

In some cases, *X* and *Y* may both be continuous random variables. For example, suppose *X* denotes the duration of an eruption (in second) of Old Faithful Geyser, and *Y* denotes the time (in minutes) until the next eruption. We might want to know if there is a relationship between *X *and *Y*. Or, we might want to know the probability that *X* falls between two particular values *a *and* b,* and *Y* falls between two particular values *c *and* d*. That is, we might want to know *P*(*a *<* X *<* b*,* c *<* Y *<* d** *).

### Objectives

- To learn the formal definition of a joint probability density function of two continuous random variables.
- To learn how to use a joint probability density function to find the probability of a specific event.
- To learn how to find a marginal probability density function of a continuous random variable
*X*from the joint probability density function of*X*and*Y*. - To learn how to find the means and variances of the continuous random variables
*X*and*Y*using their joint probability density function. - To learn the formal definition of a conditional probability density function of a continuous r.v.
*Y*given a continuous r.v.*X*. - To learn how to calculate the conditional mean and conditional variance of a continuous r.v.
*Y*given a continuous r.v.*X*. - To be able to apply the methods learned in the lesson to new problems.