##
Overview
Section* *

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) \(\text{Var}(Y|x)\), 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 \(Y\)
- 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) \(\text{Var}(Y|x)\), 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.