# Lesson 20: Distributions of Two Continuous Random Variables

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### 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 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(< 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.