Lesson 21: Bivariate Normal Distributions

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Introduction

bathroom scaleLet 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 given 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 given 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.