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4-11 WEIBULL DISTRIBUTION 133

(c) The error-correcting code might be ineffective if there are 4-104. The time between process problems in a manufac-
three or more errors within 10 5 bits. What is the probabil- turing line is exponentially distributed with a mean of 30 days.
ity of this event? (a) What is the expected time until the fourth problem?
4-102. Calls to the help line of a large computer distributor (b) What is the probability that the time until the fourth prob-
follow a Possion distribution with a mean of 20 calls per minute. lem exceeds 120 days?
(a) What is the mean time until the one-hundredth call? 4-105. Use the properties of the gamma function to evaluate
(b) What is the mean time between call numbers 50 and 80? the following:
(c) What is the probability that three or more calls occur (a) 162 (b) 15 22
within 15 seconds? (c) 19 22
4-103. The time between arrivals of customers at an auto- 4-106. Use integration by parts to show that 1r2 1r 12
matic teller machine is an exponential random variable with a 1r 12.
mean of 5 minutes.
4-107. Show that the gamma density function f 1x, , r2 in-
(a) What is the probability that more than three customers
tegrates to 1.
arrive in 10 minutes?
4-108. Use the result for the gamma distribution to determine
(b) What is the probability that the time until the fifth cus-
the mean and variance of a chi-square distribution with r 7 2.
tomer arrives is less than 15 minutes?
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).

Definition
The random variable X with probability density function
x 1 x
f 1x2 a b exp c a b d, for x 0 (4-22)

is a Weibull random variable with scale parameter 0 and shape parameter 0.

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 1 , 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.

If X has a Weibull distribution with parameters and , then the cumulative distri-
bution function of X is
x
a b
F1x2 1 e (4-23)