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PQ220 6234F.Ch 03 13/04/2002 03:19 PM Page 64

64 CHAPTER 3 DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

5X 36. Clearly, these three events are mutually exclusive. Therefore,

P1X 32 P1X 02 P1X 12 P1X 22 P1X 32
0.6561 0.2916 0.0486 0.0036 0.9999

This approach can also be used to determine

P1X 32 P1X 32 P1X 22 0.0036

Example 3-6 shows that it is sometimes useful to be able to provide cumulative proba-
bilities such as P1X x2 and that such probabilities can be used to find the probability mass
function of a random variable. Therefore, using cumulative probabilities is an alternate
method of describing the probability distribution of a random variable.
In general, for any discrete random variable with possible values x , x , p , x ,
1
2
n
the events 5X x 2, 5X x 2, p , 5X x 2 are mutually exclusive. Therefore,
1
2
n
f 1x . 2
P1X x2 g x i x i
Definition
The cumulative distribution function of a discrete random variable X, denoted as
F1x2, is
F1x2 P1X x2 a f 1x 2 i
x i x
For a discrete random variable X, F1x2 satisfies the following properties.
(1) F1x2 P1X x2 g x i x f 1x 2
i
(2) 0 F1x2 1
(3) If x y, then F1x2 F1y2 (3-2)

Like a probability mass function, a cumulative distribution function provides proba-
bilities. Notice that even if the random variable X can only assume integer values, the
cumulative distribution function can be defined at noninteger values. In Example 3-6,
F(1.5) P(X 1.5) P{X 0} P(X 1) 0.6561 0.2916 0.9477. Properties (1)
and (2) of a cumulative distribution function follow from the definition. Property (3) follows
from the fact that if x y , the event that 5X x6 is contained in the event 5X y6 .
The next example shows how the cumulative distribution function can be used to deter-
mine the probability mass function of a discrete random variable.

EXAMPLE 3-7 Determine the probability mass function of X from the following cumulative distribution
function:

0 x 2
0.2 2 x 0
F1x2 µ
0.7 0 x 2
1 2 x