Page 55 - Fundamentals of Probability and Statistics for Engineers

38 Fundamentals of Probability and Statistics for Engineers

a random variable, we assume that it is possible to assign a real number X9s)
for each outcome s following a certain set of rules. We see that the ‘number’
X9s) is really a real-valued point function defined over the domain of the basic
probability space (see Definition 3.1).
Definition 3.1. The point function X9s) is called a random variable if (a) it is a
finite real-valued function defined on the sample space S of a random experiment
for which the probability function is defined, and (b) for every real number , the
x
set fs : X9s) xg is an event. The relation X ˆ X9s) takes every element in S of
s
1
e
theprobability spaceonto a point X on thereal lin R ˆ 9 1, 1).
Notationally, the dependence of random variable X9s) on s will be omitted
for convenience.
The second condition stated in Definition 3.1 is the so-called ‘measurability
condition’. It ensures that it is meaningful to consider the probability of event
X x for every x, or, more generally, the probability of any finite or countably
infinite combination of such events.
To see more clearly the role a random variable plays in the study of a random
phenomenon, consider again the simple example where the possible outcomes
of a random experiment are success and failure. Let us again assign number one
to the event success and zero to failure. If X is the random variable associated
with this experiment, then X takes on two possible values: 1 and 0. Moreover,
the following statements are equivalent:
. X ˆ 1.
. The outcome is success.
. The outcome is 1.

The random variable X is called a discrete random variable if it is defined
over a sample space having a finite or a countably infinite number of sample
points. In this case, random variable X takes on discrete values, and it is
possible to enumerate all the values it may assume. In the case of a sample
space having an uncountably infinite number of sample points, the associated
random variable is called a continuous random variable, with its values dis-
tributed over one or more continuous intervals on the real line. We make this
distinction because they require different probability assignment consider-
ations. Both types of random variables are important in science and engineering
and we shall see ample evidence of this in the subsequent chapters.
In the following, all random variables will be written in capital letters,
X, Y, Z ,.... The value that a random variable X can assume will be denoted
by corresponding lower-case letters such as x, y, z , or x 1 , x 2 ,....
We will have many occasions to consider a sequence of random variables
X j , j ˆ 1, 2, ... , n. In these cases we assume that they are defined on the same
probability space. The random variables X 1 , X 2 , ... , X n will then map every
element s of S in the probability space onto a point in the -dimensional
n

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