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Chapter 3

Multiple Random Variables

3.1 INTRODUCTION
In many applications it is important to study two or more r.v.'s defined on the same sample
space. In this chapter, we first consider the case of two r.v.'s, their associated distribution, and some
properties, such as independence of the r.v.'s. These concepts are then extended to the case of many
r.v.'s defined on the same sample space.

3.2. BIVARIATE RANDOM VARIABLES
A. Definition:

Let S be the sample space of a random experiment. Let X and Y be two r.v.'s. Then the pair (X,
Y) is called a bivariate r.v. (or two-dimensional random vector) if each of X and Y associates a real
number with every element of S. Thus, the bivariate r.v. (X, Y) can be considered as a function that to
each point c in S assigns a point (x, y) in the plane (Fig. 3-1). The range space of the bivariate r.v. (X,
Y) is denoted by R,, and defined by

If the r.v.'s X and Y are each, by themselves, discrete r.v.'s, then (X, Y) is called a discrete
bivariate r.v. Similarly, if X and Y are each, by themselves, continuous r.v.'s, then (X, Y) is called a
continuous bivariate r.v. If one of X and Y is discrete while the other is continuous, then (X, Y) is
called a mixed bivariate r.v.

Fig. 3-1 (X, Y) as a function from S to the plane.
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