Page 46 - Probability, Random Variables and Random Processes

Chapter 2

2.1 INTRODUCTION
In this chapter, the concept of a random variable is introduced. The main purpose of using a
random variable is so that we can define certain probability functions that make it both convenient
and easy to compute the probabilities of various events.

2.2 RANDOM VARIABLES
A. Definitions:
Consider a random experiment with sample space S. A random variable X(c) is a single-valued
real function that assigns a real number called the value of X([) to each sample point [ of S. Often, we
use a single letter X for this function in place of X(5) and use r.v. to denote the random variable.
Note that the terminology used here is traditional. Clearly a random variable is not a variable at
all in the usual sense, and it is a function.
The sample space S is termed the domain of the r.v. X, and the collection of all numbers [values
of X([)] is termed the range of the r.v. X. Thus the range of X is a certain subset of the set of all real
numbers (Fig. 2-1).
Note that two or more different sample points might give the same value of X(0, but two differ-
ent numbers in the range cannot be assigned to the same sample point.

x (0 R
Fig. 2-1 Random variable X as a function.

EXAMPLE 2.1 In the experiment of tossing a coin once (Example 1.1), we might define the r.v. X as (Fig. 2-2)
X(H) = 1 X(T) = 0

Note that we could also define another r.v., say Y or 2, with
Y(H) = 0, Y(T) = 1 or Z(H) = 0, Z(T) = 0

B. Events Defined by Random Variables:
If X is a r.v. and x is a fixed real number, we can define the event (X = x) as
(X = x) = {l: X(C) = x)
Similarly, for fixed numbers x, x,, and x, , we can define the following events:
(X 5 x) = {l: X(l) I x)
(X > x) = {C: X([) > x)
(xl < X I x2) = {C: XI < X(C) l x2)