Page 149 - Applied Statistics And Probability For Engineers

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4-9 EXPONENTIAL DISTRIBUTION 127

memory property of the exponential distribution implies that the device does not wear out.
That is, regardless of how long the device has been operating, the probability of a failure
in the next 1000 hours is the same as the probability of a failure in the first 1000 hours of
operation. The lifetime L of a device with failures caused by random shocks might be ap-
propriately modeled as an exponential random variable. However, the lifetime L of a
device that suffers slow mechanical wear, such as bearing wear, is better modeled by a dis-
tribution such that P1L t
t 0 L t2 increases with t. Distributions such as the Weibull
distribution are often used, in practice, to model the failure time of this type of device. The
Weibull distribution is presented in a later section.

EXERCISES FOR SECTION 4-9
4-72. Suppose X has an exponential distribution with 2. (b) What is the probability that at least one call arrives within
Determine the following: a 10-minute interval?
(a) P1X 02 (b) P1X 22 (c) What is the probability that the first call arrives within 5
(c) P1X 12 (d) P11 X 22 and 10 minutes after opening?
(e) Find the value of x such that P1X x2 0.05. (d) Determine the length of an interval of time such that the
probability of at least one call in the interval is 0.90.
4-73. Suppose X has an exponential distribution with mean
4-78. The life of automobile voltage regulators has an expo-
equal to 10. Determine the following:
nential distribution with a mean life of six years. You purchase
(a) P1X 102
an automobile that is six years old, with a working voltage
(b) P1X 202
regulator, and plan to own it for six years.
(c) P1X 302
(a) What is the probability that the voltage regulator fails dur-
(d) Find the value of x such that P1X x2 0.95.
ing your ownership?
4-74. Suppose the counts recorded by a geiger counter follow (b) If your regulator fails after you own the automobile three
a Poisson process with an average of two counts per minute. years and it is replaced, what is the mean time until the
(a) What is the probability that there are no counts in a 30- next failure?
second interval?
4-79. The time to failure (in hours) of fans in a personal com-
(b) What is the probability that the first count occurs in less puter can be modeled by an exponential distribution with
than 10 seconds? 0.0003.
(c) What is the probability that the first count occurs between (a) What proportion of the fans will last at least 10,000 hours?
1 and 2 minutes after start-up? (b) What proportion of the fans will last at most 7000 hours?
4-75. Suppose that the log-ons to a computer network fol- 4-80. The time between the arrival of electronic messages at
low a Poisson process with an average of 3 counts per minute. your computer is exponentially distributed with a mean of two
(a) What is the mean time between counts? hours.
(b) What is the standard deviation of the time between counts? (a) What is the probability that you do not receive a message
(c) Determine x such that the probability that at least one during a two-hour period?
count occurs before time x minutes is 0.95. (b) If you have not had a message in the last four hours, what
4-76. The time to failure (in hours) for a laser in a cytome- is the probability that you do not receive a message in the
try machine is modeled by an exponential distribution with next two hours?
0.00004. (c) What is the expected time between your fifth and sixth
(a) What is the probability that the laser will last at least messages?
20,000 hours? 4-81. The time between arrivals of taxis at a busy intersec-
(b) What is the probability that the laser will last at most tion is exponentially distributed with a mean of 10 minutes.
30,000 hours? (a) What is the probability that you wait longer than one hour
(c) What is the probability that the laser will last between for a taxi?
20,000 and 30,000 hours? (b) Suppose you have already been waiting for one hour for a
4-77. The time between calls to a plumbing supply business taxi, what is the probability that one arrives within the
is exponentially distributed with a mean time between calls of next 10 minutes?
15 minutes. 4-82. Continuation of Exercise 4-81.
(a) What is the probability that there are no calls within a 30- (a) Determine x such that the probability that you wait more
minute interval? than x minutes is 0.10.