Applied Statistics and Probability for Engineers

64 CHAPTER 3 DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

Clearly, these three events are mutually exclusive. Therefore,

This approach can also be used to determine

Example 3-6 shows that it is sometimes useful to be able to provide cumulative proba-

bilitiessuch as 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

the events are mutually exclusive. Therefore,

P 1 Xx 2 gxix f 1 xi 2.

5 Xx 12 , 5 Xx 22 ,p, 5 Xxn 2

x 1 , x 2 ,p, xn,

P 1 Xx 2

P 1 X 32 P 1 X 32 P 1 X 22 0.0036

0.6561 0.2916 0.0486 0.00360.9999

P 1 X 32 P 1 X 02     P 1 X 12   P 1 X 22   P 1 X 32

5 X 36.

The cumulative distribution function of a discrete random variable X, denoted as

is

For a discrete random variable X, satisfies the following properties.

(1)

(2)

(3) If xy, then F 1 x 2 F 1 y 2 (3-2)

0 F 1 x 2  1

F 1 x 2 P 1 Xx 2  gxix f 1 xi 2

F 1 x 2

F 1 x 2 P 1 Xx 2  a

xix

f 1 xi 2

F 1 x 2 ,

Definition

Like a probability mass function, a cumulative distribution function provides proba-

bilities. Notice that even if the random variable Xcan 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{X0} 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 , the event that is contained in the event.

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 Xfrom the following cumulative distribution

function:

F 1 x 2 μ

0 x 2

0.2  2 x 0

0.7 0 x 2

12 x

xy 5 Xx 6 5 Xy 6

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