Lesson 21: Bivariate Normal Distributions

Overview Section

bathroom scale

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

Upon completion of this lesson, you should be able to:

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