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B–5 Cumulative Distribution Functions and Probability Density Functions 685
while B was the event that it was raining at the intersection, then A and B would be independent.
Why?
Using Eqs. (B–8) and (B–10), we can show that if a set of events A 1 , A 2 ,..., A n , are
independent, then a necessary condition is †
P(A 1 A 2 Á A n ) = P(A 1 )P(A 2 ) Á P(A n )
(B–12)
B–4 RANDOM VARIABLES
DEFINITION. A real-valued random variable is a real-valued function defined on the
events (elements) of the probability system.
An understanding of why this definition is needed is fundamental to the topic of probability
theory. So far, we have defined probabilities in terms of events A, B, C, and so on. This
method is awkward to use when the sets are objects (apples, oranges, etc.) instead of numbers.
It is more convenient to describe sets by numerical values, so that equations can be obtained
as a function of numerical values instead of functions of alphanumeric parameters. This
method is accomplished by using the random variable.
Example B–2 RANDOM VARIABLE
Referring to Fig. B–2, we can show the mutually exclusive events A, B, C, D, E, F, G, and H
by a Venn diagram. These are all the possible outcomes of an experiment, so the sure event is
S = A + B + C + D + E + F + G + H. Each of these events is denoted by some convenient value of
the random variable x, as shown in the table in this figure. The assigned values for x may be
positive, negative, fractions, or integers as long as they are real numbers. Since all the events are
mutually exclusive, using (B–4) yields
P(S) = 1 = P(A) + P(B) + P(C) + P(D) + P(E) + P(F) + P(G) + P(H)
(B–13)
That is, the probabilities have to sum to unity (the probability of a sure event), as shown in
the table, and the individual probabilities have been given or measured as shown. For example,
P(C) = P(-1.5) = 0.2. These values for the probabilities may be plotted as a function of the
random variable x, as shown in the graph of P(x). This is a discrete (or point) distribution since
the random variable takes on only discrete (as opposed to continuous) values.
B–5 CUMULATIVE DISTRIBUTION FUNCTIONS
AND PROBABILITYDENSITY FUNCTIONS
DEFINITION. The cumulative distribution function (CDF) of the random variable x is
given by F(a), where
where F(a) is a unitless function.
F(a) ! P(x … a) K lim
an x…a
n: q n
b
(B–14)
† Equation (B–12) is not a sufficient condition for A 1 , A 2 ,..., A n to be independent [Papoulis, 1984, p. 34].