Unformatted text preview: 140 4—78. Suppose X has an exponential distribution with a mean of 10. Determine the following: (a) P(X < 5) (b) P(X< 15X> 10) (c) Compare the results in parts (a) and (b) and comment on
the role of the lack of memory property. 4—79. Suppose the counts recorded by a Geiger counter follow a Poisson process with an average of two counts per minute. (a) What is the probability that there are no counts in a 30
second interval? (b) What is the probability that the ﬁrst count occurs in less
than 10 seconds? (c) What is the probability that the ﬁrst count occurs between
1 and 2 minutes after startup? 4—80. Suppose that the logons to a computer network fol low a Poisson process with an average of 3 counts per minute. (a) What is the mean time between counts? (b) What is the standard deviation of the time between counts? (c) Determine x such that the probability that at least one
count occurs before time x minutes is 0.95. 4—81. The time between calls to a plumbing supply business is exponentially distributed with a mean time between calls of 15 minutes. (a) What is the probability that there are no calls within a 30
minute interval? (b) What is the probability that at least one call arrives within
a 10minute interval? (c) What is the probability that the ﬁrst call arrives within 5
and 10 minutes after opening? ((1) Determine the length of an interval of time such that the
probability of at least one call in the interval is 0.90. 4—82. The life of automobile voltage regulators has an expo nential distribution with a mean life of six years. You purchase an automobile that is six years old, with a working voltage regulator, and plan to own it for six years. (a) What is the probability that the voltage regulator fails dur
ing your ownership? (b) If your regulator fails after you own the automobile three
years and it is replaced, what is the mean time until the
next failure? 4—83. Suppose that the time to failure (in hours) of fans in a
personal computer can be modeled by an exponential distribu—
tion with )t = 0.0003. (a) What proportion of the fans will last at least 10,000 hours?
(b) What proportion of the fans will last at most 7000 hours? 4—84. The time between the arrival of electronic messages at your computer is exponentially distributed with a mean of two hours. (a) What is the probability that you do not receive a message
during a twohour period? (b) If you have not had a message in the last four hours, what
is the probability that you do not receive a message in the
next two hours? CHAPTER 4 CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS (c) What is the expected time between your ﬁfth and six
messages? 4—85. The time between arrivals of taxis at a busy intersec' tion is exponentially distributed with a mean of 10 minute (a) What is the probability that you wait longer than one ho
for a taxi? " (b) Suppose you have already been waiting for one hour for
taxi, what is the probability that one arrives within th
next 10 minutes? (c) Determine x such that the probability that you wait mo
than x minutes is 0.10. (d) Determine x such that the probability that you wait 1e
than x minutes is 0.90. (e) Determine x such that the probability that you wait le
than x minutes is 0.50. 4—86. The number of stork sightings on a route in Sout : Carolina follows a Poisson process with a mean of 2.3 per year. (a) What is the mean time between sightings? (b) What is the probability that there are no sightings wi '
three months (0.25 years)? _ (c) What is the probability that the time until the ﬁrst sightinga
exceeds six months? (d) What is the probability of no sighting within three years? 4—87. According to results from the analysis of chocolate bars in Chapter 3, the mean number of insect fragments was 14.4 in 225 grams. Assume the number of fragments follow a Poisson; distribution. (a) What is the mean number of grams of chocolate until a
fragment is detected? (b) What is the probability that there are no fragments in a
28.35 gram (one ounce) chocolate bar? (c) Suppose you consume seven oneounce (28.35 grams) bars
this week. What is the probability of no insect fragments? 4—88. The distance between major cracks in a highway fol— lows an exponential distribution with a mean of 5 miles. (a) What is the probability that there are no major cracks in a .
10mile stretch of the highway? (b) What is the probability that there are two major cracks in
a 10mile stretch of the highway? (c) What is the standard deviation of the distance between
major cracks? L (d) What is the probability that the ﬁrst major crack occurs
between 12 and 15 miles of the start of inspection? (e) What is the probability that there are no major cracks in
two separate 5mile stretches of the highway? (f) Given that there are no cracks in the ﬁrst 5 miles in ‘
spected, what is the probability that there are no major 5
cracks in the next 10 miles inspected? 4—89. The lifetime of a mechanical assembly in a vibration i
test is exponentially distributed with a mean of 400 hours. .
(a) What is the probability that an assembly on test fails in 1 less than 100 hours? I ...
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 Spring '08
 Wherly
 Statistics

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