# Lesson 21: Bivariate Normal Distributions

### Introduction

Let the random variable *Y *denote the weight of a randomly selected individual, in pounds. Then, suppose we are interested in determining the probability that a randomly selected individual weighs between 140 and 160 pounds. That is, what is *P*(140 < *Y *< 160)?

But, if we think about it, we could imagine that the weight of an individual increases (linearly?) as height increases. If that's the case, in calculating the probability that a randomly selected individual weighs between 140 and 160 pounds, we might find it more informative to first take into account a person's height, say *X*. That is, we might want to find instead *P*(140 < *Y *< 160| *X *= *x*). To calculate such a conditional probability, we clearly first need to find the conditional distribution of *Y *given *X *= *x*. That's what we'll do in this lesson, that is, after first making a few assumptions.

First, we'll assume that (1) *Y* follows a normal distribution, (2) *E*(*Y*|*x*), the conditional mean of *Y* given *x* is linear in *x*, and (3) *Var*(*Y*|*x*), the conditional variance of *Y* given *x* is constant. Based on these three stated assumptions, we'll find the conditional distribution of *Y *given *X *= *x*.

Then, to the three assumptions we've already made, we'll then add the assumption that the random variable *X* follows a normal distribution, too. Based on the now four stated assumptions, we'll find the joint probability density function of *X* and *Y*.

### Objectives

- To find the conditional distribution of
*Y*given*X*=*x*, assuming that (1)*Y*follows a normal distribution, (2)*E*(*Y*|*x*), the conditional mean of*Y*given*x*is linear in*x*, and (3)*Var*(*Y*|*x*), the conditional variance of*Y*given*x*is constant. - To learn how to calculate conditional probabilities using the resulting conditional distribution.
- To find the joint distribution of
*X*and*Y,*assuming that (1)*X*follows a normal distribution, (2)*Y*follows a normal distribution, (3)*E*(*Y*|*x*), the conditional mean of*Y*given*x*is linear in*x*, and (4)*Var*(*Y*|*x*), the conditional variance of*Y*given*x*is constant. - To learn the formal definition of the bivariate normal distribution.
- To understand that when
*X*and*Y*have the bivariate normal distribution with zero correlation, then*X*and*Y*must be independent. - To understand each of the proofs provided in the lesson.
- To be able to apply the methods learned in the lesson to new problems.