Applied Statistics and Probability for Engineers

3-3 CUMULATIVE DISTRIBUTION FUNCTIONS 65

Figure 3-3 displays a plot of From the plot, the only points that receive nonzero

probability are 2, 0, and 2. The probability mass function at each point is the change in the

cumulative distribution function at the point. Therefore,

EXAMPLE 3-8 Suppose that a day’s production of 850 manufactured parts contains 50 parts that do not con-

form to customer requirements. Two parts are selected at random, without replacement, from

the batch. Let the random variable Xequal the number of nonconforming parts in the sample.

What is the cumulative distribution function of X?

The question can be answered by first finding the probability mass function of X.

Therefore,

The cumulative distribution function for this example is graphed in Fig. 3-4. Note that

is defined for all xfrom and not only for 0, 1, and 2.

EXERCISES FOR SECTION 3-3

F 1 x 2 

x

F 122 P 1 X 22  1

F 112 P 1 X 12 0.886 0.1110.997

F 102 P 1 X 02 0.886

P 1 X 22 

50

850



49

849

0.003

P 1 X 12  2 

800

850



50

849

0.111

P 1 X 02 

800

850



799

849

0.886

f 1  22 0.2 0 0.2 f 102 0.70.20.5 f 122 1.00.70.3

F 1 x 2.

0

0.2

–2 2

0.7

1.0

x

F(x)

Figure 3-3 Cumulative distribution function for

Example 3-7.

Figure 3-4 Cumulative distribution

function for Example 3-8.

0 2

0.997

1.000

x

0.886

1

F(x)

3-26. Determine the cumulative distribution function of the

random variable in Exercise 3-13.

3-27. Determine the cumulative distribution function for

the random variable in Exercise 3-15; also determine the fol-

lowing probabilities:

(a)P 1 X1.25 2 (b)P 1 X2.2 2

(c) (d)

3-28. Determine the cumulative distribution function for the

random variable in Exercise 3-17; also determine the following

probabilities:

(a) (b)

(c)P 1 X 22 (d)P 11 X 22

P 1 X1.5 2 P 1 X 32

P 1 1.1X 12 P 1 X 02

PQ220 6234F.Ch 03 13/04/2002 03:19 PM Page 65