B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

408 FUNDAMENTALS OF PROBABILITY THEORY
then, based upon these axioms alone, what else is true? As Bertrand Russell puts it: "Pure mathematics
consists entirely of such asseverations as that. if such and such proposition is true of
anything, then such and such another proposition is true of that thing." Seen in this light, it may
appear that assigning probability to an event may not necessarily be the responsibility of the
mathematical discipline of probability. Under mathematical discipline, we need to start with
a set of axioms about probability and then investigate what else can be said about probability
based on this set of axioms alone. We start with a concept (as yet undefined) of probability
and postulate axioms. The axioms must be internally consistent and should conform to the
observed relationships and behavior of probability in the practical and the intuitive sense. It
is beyond the scope of this book to discuss how these axioms are formulated. The modern
theory of probability starts with Eqs. (8.6), (8.8), and (8.11) as its axioms. Based on these three
axioms alone, what else is true is the essence of modern theory of probability. The relative
frequency approach uses Eq. (8.5) to define probability, and Eqs. (8.5), (8.8), and (8. 11) follow
as a consequence of this definition. In the axiomatic approach, on the other hand, we do not
say anything about how we assign probability P(A) to an event A; rather, we postulate that the
probability function must obey the three postulates or axioms in Eqs. (8.6), (8.8), and (8.11 ).
The modem theory of probability does not concern itself with the problem of assigning probabilities
to events. It assumes that somehow the probabilities were assigned to these events a
priori.
If a mathematical model is to conform to the real phenomenon, we must assign these
probabilities in away that is consistent with an empirical and an intuitive understanding of
probability. The concept of relative frequency is admirably suited for this. Thus, although we
use relative frequency to assign (not define) probabilities, it is all under the table, not a part of
the mathematical discipline of probability.
8.2 RANDOM VARIABLES
The outcome of an experiment may be a real number (as in the case of rolling a die), or it
may be nonnumerical and describable by a phrase (such as "heads" or "tail" in tossing a coin).
From a mathematical point of view, it is simpler to have numerical values for all outcomes.
For this reason, we assign a real number to each sample point according to some rule. If there
are m sample points (1 , (2, ... , t m , then using some convenient rule, we assign a real number
x((i) to sample point (i (i = 1, 2, ... , m). In the case of tossing a coin, for example, we may
assign the number 1 for the outcome heads and the number - 1 for the outcome tails (Fig. 8.5).
Thus, x(.) is a function that maps sample points (1 , (2, ... , t m into real numbers
x1 , x2, ... , X n .* We now have a random variable x that takes on values x1 , x2, ... , X n ,
We shall use roman type (x) to denote a random variable (RV) and italic type (e.g.,
x1 , x2, ... , X n ) to denote the value it takes. The probability of an RV x taking a value Xi
is P x (Xi) = Probability of "x = Xi."
Discrete Random Variables
A random variable is discrete if there exists a denumerable sequence of distinct numbers Xi
such that
(8. 19)
* The number m is not necessarily equal to n. More than one sample point can map into one value of x.