Page 159 - Applied statistics and probability for engineers

Section 4-8/Exponential Distribution 137

The exponential distribution is often used in reliability studies as the model for the time until
failure of a device. For example, the lifetime of a semiconductor chip might be modeled as an
exponential random variable with a mean of 40,000 hours. The lack of 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 irst 1000 hours of operation. The lifetime L of a device with failures
caused by random shocks might be appropriately 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 distribution such that P L < t + Δ u
t L > t) increases with t. Distri-
butions 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 4-8

Problem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion.
4-112. Suppose that X has an exponential distribution (c) What is the probability that the irst call arrives within 5
with λ = 2. Determine the following: and 10 minutes after opening?
(
(
(a) P X ≤ ) 0 (b) P X ≥ ) 2 (d) Determine the length of an interval of time such that the
(
1 (
(c) P X ≤ ) 1 (d) P < X < 2) probability of at least one call in the interval is 0.90.
(
(e) Find the value of x such that P X < x) = .05 . 4-118. The life of automobile voltage regulators has
0
an exponential distribution with a mean life of six years. You
4-113. Suppose that X has an exponential distribution
purchase a six-year-old automobile, with a working voltage
with mean equal to 10. Determine the following:
(a) P X >10) (b) P X > 20) (c) P X <30) regulator and plan to own it for six years.
(
(
(
(
(d) Find the value of x such that P X < x) = .95 . (a) What is the probability that the voltage regulator fails dur-
0
ing your ownership?
4-114. Suppose that X has an exponential distribution
(b) If your regulator fails after you own the automobile three years
with a mean of 10. Determine the following: and it is replaced, what is the mean time until the next failure?
(
(a) P X < 5) (b) P X <15| X >10)
(
(c) Compare the results in parts (a) and (b) and comment on 4-119. Suppose that the time to failure (in hours) of fans
in a personal computer can be modeled by an exponential dis-
the role of the lack of memory property.
tribution with λ = .0 0003 .
4-115. Suppose that the counts recorded by a Geiger (a) What proportion of the fans will last at least 10,000 hours?
counter follow a Poisson process with an average of two counts (b) What proportion of the fans will last at most 7000 hours?
per minute. 4-120. The time between the arrival of electronic mes-
(a) What is the probability that there are no counts in a 30-sec- sages at your computer is exponentially distributed with a mean
ond interval? of two hours.
(b) What is the probability that the irst count occurs in less (a) What is the probability that you do not receive a message
than 10 seconds? during a two-hour period?
(c) What is the probability that the irst count occurs between (b) If you have not had a message in the last four hours, what
one and two minutes after start-up? is the probability that you do not receive a message in the
4-116. Suppose that the log-ons to a computer network next two hours?
follow a Poisson process with an average of three counts per (c) What is the expected time between your ifth and sixth
minute. messages?
(a) What is the mean time between counts? 4-121. The time between arrivals of taxis at a busy inter-
(b) What is the standard deviation of the time between counts? section is exponentially distributed with a mean of 10 minutes.
(c) Determine x such that the probability that at least one count (a) What is the probability that you wait longer than one hour
occurs before time x minutes is 0.95. for a taxi?
4-117. The time between calls to a plumbing supply busi- (b) Suppose that you have already been waiting for one hour
ness is exponentially distributed with a mean time between for a taxi. What is the probability that one arrives within the
calls of 15 minutes. next 10 minutes?
(a) What is the probability that there are no calls within a (c) Determine x such that the probability that you wait more
30-minute interval? than x minutes is 0.10.
(b) What is the probability that at least one call arrives within (d) Determine x such that the probability that you wait less
a 10-minute interval? than x minutes is 0.90.