Probability, Statistics, and Stochastic Processes, 2nd Edition

Chapter 3

Joint Distributions

3.1 Introduction

In the previous chapter, we introduced random variables to describe random experiments with numerical outcomes. We restricted our attention to cases where the outcome is a single number, but there are many cases where the outcome is a vector of numbers. We have already seen one such experiment, in Example 1.5, where a dart is thrown at random on a dartboard of radius r. The outcome is a pair (X, Y) of random variables that are such that X² + Y² ≤ r². For another example, suppose that we measure voltage and current in an electric circuit with known resistance. Owing to random fluctuations and measurement error, we can view this as an outcome (V, I) of a pair of random variables.

These examples have in common that there is a relation between the random variables that we measure, and by describing them only one by one, we do not get all the possible information. The dart coordinates are restricted by the board, and voltage and current are related by Ohm's law. In this chapter, we extend the notion of random variables to random vectors.

3.2 The Joint Distribution Function

We will focus primarily on random vectors in two dimensions. Let us begin by formally defining our fundamental object of study.

Definition 3.1. Let X and Y be random variables. The pair (X, Y) is then called a (two-dimensional) random vector.

Our aim is to describe random vectors in much the same way as we described random variables in Chapter 2, and we will thus ...