Applied Statistics and Probability for Engineers

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 Xconsists of three expressions. If Therefore,

F 1 x 2 0, for x 0

x0, f 1 x 2 0.

102 CHAPTER 4 CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

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.

The cumulative distribution functionof a continuous random variable Xis

(4-3)

for x.

F 1 x 2 P 1 Xx 2  

x



f 1 u 2 du

Definition

(c) (d)

(e) Determine xsuch that P(Xx) 0.90.

4-5. Suppose that for Determine

the following probabilities:

(a) (b)

(c) (d)

(e)

(f) Determine xsuch that

4-6. The probability density function of the time to failure

of an electronic component in a copier (in hours) is f(x) 

for Determine the probability that

(a) A component lasts more than 3000 hours before failure.

(b) A component fails in the interval from 1000 to 2000 hours.

(c) A component fails before 1000 hours.

(d) Determine the number of hours at which 10% of all com-

ponents have failed.

4-7. The probability density function of the net weight in

pounds of a packaged chemical herbicide is for

pounds.

(a) Determine the probability that a package weighs more

than 50 pounds.

49.75x50.25

f 1 x 2 2.0

x0.

ex^1000

1000

P 1 xX 2 0.05.

P 1 X0 or X0.5 2

P 1 0.5X0.5 2 P 1 X 22

P 10 X 2 P 1 0.5X 2

f 1 x 2 1.5x^2  1 x1.

P 15 X 2 P 18 X 122 (b) How much chemical is contained in 90% of all packages?

4-8. The probability density function of the length of a

hinge for fastening a door is for

millimeters. Determine the following:

(a)

(b)

(c) If the specifications for this process are from 74.7

to 75.3 millimeters, what proportion of hinges meets

specifications?

4-9. The probability density function of the length of a

metal rod is for 2.3 x2.8 meters.

(a) If the specifications for this process are from 2.25 to 2.75

meters, what proportion of the bars fail to meet the speci-

fications?

(b) Assume that the probability density function is

for an interval of length 0.5 meters. Over what value

should the density be centered to achieve the greatest pro-

portion of bars within specifications?

4-10. If Xis a continuous random variable, argue that P(x 1 

Xx 2 ) P(x 1 Xx 2 ) P(x 1 Xx 2 ) P(x 1 Xx 2 ).

f 1 x 2  2

f 1 x 2  2

P 1 X74.8 or X75.2 2

P 1 X74.8 2

f 1 x 2 1.25 74.6x75.4

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