Page 124 - Applied Statistics And Probability For Engineers

c04.qxd 5/10/02 5:19 PM Page 102 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH114 FIN L:Quark Files:

102 CHAPTER 4 CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

(c) P15 X 2 (d) P18 X 122 (b) How much chemical is contained in 90% of all packages?
(e) Determine x such that P(X x) 0.90. 4-8. The probability density function of the length of a
4-5. Suppose that f 1x2 1.5x 2 for 1 x 1. Determine hinge for fastening a door is f 1x2 1.25 for 74.6 x 75.4
the following probabilities: millimeters. Determine the following:
(a) P10 X 2 (b) P10.5 X 2 (a) P1X 74.82
(c) P1 0.5 X 0.52 (d) P1X 22 (b) P1X 74.8 or X 75.22
(e) P1X 0 or X 0.52 (c) If the specifications for this process are from 74.7
(f) Determine x such that P1x X 2 0.05. to 75.3 millimeters, what proportion of hinges meets
4-6. The probability density function of the time to failure specifications?
of an electronic component in a copier (in hours) is f(x) 4-9. The probability density function of the length of a
e x 1000 metal rod is f 1x2 2 for 2.3 x 2.8 meters.
for x 0. Determine the probability that
1000 (a) If the specifications for this process are from 2.25 to 2.75
(a) A component lasts more than 3000 hours before failure. meters, what proportion of the bars fail to meet the speci-
(b) A component fails in the interval from 1000 to 2000 hours. fications?
(c) A component fails before 1000 hours. (b) Assume that the probability density function is f 1x2 2
(d) Determine the number of hours at which 10% of all com- for an interval of length 0.5 meters. Over what value
ponents have failed. should the density be centered to achieve the greatest pro-
portion of bars within specifications?
4-7. The probability density function of the net weight in
pounds of a packaged chemical herbicide is f 1x2 2.0 for 4-10. If X is a continuous random variable, argue that P(x 1
49.75 x 50.25 pounds. X x 2 ) P(x 1 X x 2 ) P(x 1 X x 2 ) P(x 1 X x 2 ).
(a) Determine the probability that a package weighs more
than 50 pounds.

4-3 CUMULATIVE DISTRIBUTION FUNCTIONS

An alternative method to describe the distribution of a discrete random variable can also be
used for continuous random variables.

Definition
The cumulative distribution function of a continuous random variable X is
x

F1x2 P1X x2 f 1u2 du (4-3)

for x .

Extending the definition of f(x) to the entire real line enables us to define the cumulative dis-
tribution function for all real numbers. The following example illustrates the definition.

EXAMPLE 4-3 For the copper current measurement in Example 4-1, the cumulative distribution function of
the random variable X consists of three expressions. If x 0, f 1x2 0. Therefore,

F1x2 0, for x 0