Applied Statistics and Probability for Engineers

4-11 WEIBULL DISTRIBUTION 133

(c) The error-correcting code might be ineffective if there are

three or more errors within bits. What is the probabil-

ity of this event?

4-102. Calls to the help line of a large computer distributor

follow a Possion distribution with a mean of 20 calls per minute.

(a) What is the mean time until the one-hundredth call?

(b) What is the mean time between call numbers 50 and 80?

(c) What is the probability that three or more calls occur

within 15 seconds?

4-103. The time between arrivals of customers at an auto-

matic teller machine is an exponential random variable with a

mean of 5 minutes.

(a) What is the probability that more than three customers

arrive in 10 minutes?

(b) What is the probability that the time until the fifth cus-

tomer arrives is less than 15 minutes?

105

4-104. The time between process problems in a manufac-

turing line is exponentially distributed with a mean of 30 days.

(a) What is the expected time until the fourth problem?

(b) What is the probability that the time until the fourth prob-

lem exceeds 120 days?

4-105. Use the properties of the gamma function to evaluate

the following:

(a) (b)

(c)

4-106. Use integration by parts to show that

4-107. Show that the gamma density function in-

tegrates to 1.

4-108. Use the result for the gamma distribution to determine

the mean and variance of a chi-square distribution with r 7 2.

f 1 x, , r 2

 1 r 12.

 1 r 2  1 r 12

 (^19)  22
 162  (^15)  22
4-11 WEIBULL DISTRIBUTION
As mentioned previously, the Weibull distribution is often used to model the time until failure
of many different physical systems. The parameters in the distribution provide a great deal of
flexibility to model systems in which the number of failures increases with time (bearing
wear), decreases with time (some semiconductors), or remains constant with time (failures
caused by external shocks to the system).
The flexibility of the Weibull distribution is illustrated by the graphs of selected probability
density functions in Fig. 4-27. By inspecting the probability density function, it is seen that
when , the Weibull distribution is identical to the exponential distribution.
The cumulative distribution function is often used to compute probabilities. The follow-
ing result can be obtained.
 1
The random variable Xwith probability density function
(4-22)
is a Weibull random variablewith scale parameter 0 and shape parameter 0.
f 1 x 2 


a
x

b
 1

exp ca

x



b



d, for x 0

Definition

If Xhas a Weibull distribution with parameters and , then the cumulative distri-

bution function ofXis

F 1 x 2  1 e (4-23)

a

x

b



 

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