Page 160 - Applied statistics and probability for engineers

138 Chapter 4/Continuous Random Variables and Probability Distributions

(e) Determine x such that the probability that you wait less 4-127. The time between calls to a corporate ofice is expo-
than x minutes is 0.50. nentially distributed with a mean of 10 minutes.
4-122. The number of stork sightings on a route in South (a) What is the probability that there are more than three calls
Carolina follows a Poisson process with a mean of 2.3 per year. in one-half hour?
(a) What is the mean time between sightings? (b) What is the probability that there are no calls within one-
(b) What is the probability that there are no sightings within half hour?
three months (0.25 years)? (c) Determine x such that the probability that there are no calls
(c) What is the probability that the time until the irst sighting within x hours is 0.01.
exceeds six months? (d) What is the probability that there are no calls within a two-
(d) What is the probability of no sighting within three years? hour interval?
4-123. According to results from the analysis of chocolate bars (e) If four nonoverlapping one-half-hour intervals are selected,
in Chapter 3, the mean number of insect fragments was 14.4 what is the probability that none of these intervals contains
in 225 grams. Assume that the number of fragments follows a any call?
Poisson distribution. (f) Explain the relationship between the results in part (a) and (b).
(a) What is the mean number of grams of chocolate until a
4-128. Assume that the laws along a magnetic tape fol-
fragment is detected?
low a Poisson distribution with a mean of 0.2 law per meter.
(b) What is the probability that there are no fragments in a
Let Xdenote the distance between two successive laws.
28.35-gram (one-ounce) chocolate bar?
(a) What is the mean of X?
(c) Suppose you consume seven one-ounce (28.35-gram) bars
(b) What is the probability that there are no laws in 10 con-
this week. What is the probability of no insect fragments?
secutive meters of tape?
4-124. The distance between major cracks in a highway
(c) Does your answer to part (b) change if the 10 meters are
follows an exponential distribution with a mean of ive miles.
not consecutive?
(a) What is the probability that there are no major cracks in a
(d) How many meters of tape need to be inspected so that the
10-mile stretch of the highway?
probability that at least one law is found is 90%?
(b) What is the probability that there are two major cracks in a
(e) What is the probability that the irst time the distance
10-mile stretch of the highway?
between two laws exceeds eight meters is at the ifth law?
(c) What is the standard deviation of the distance between
(f) What is the mean number of laws before a distance between
major cracks?
two laws exceeds eight meters?
(d) What is the probability that the irst major crack occurs
between 12 and 15 miles of the start of inspection? 4-129. If the random variable X has an exponential distribu-
(e) What is the probability that there are no major cracks in tion with mean θ, determine the following: (
(c) P X >3θ)
(
(a) P X >θ)
(
(b) P X > 2θ)
two separate ive-mile stretches of the highway?
(f) Given that there are no cracks in the irst ive miles (d) How do the results depend on θ?
inspected, what is the probability that there are no major 4-130. Derive the formula for the mean and variance of
cracks in the next 10 miles inspected? an exponential random variable.
4-125. The lifetime of a mechanical assembly in a vibra- 4-131. Web crawlers need to estimate the frequency of changes
tion test is exponentially distributed with a mean of 400 hours. to Web sites to maintain a current index for Web searches.
(a) What is the probability that an assembly on test fails in less Assume that the changes to a Web site follow a Poisson process
than 100 hours? with a mean of 3.5 days.
(b) What is the probability that an assembly operates for more (a) What is the probability that the next change occurs in less
than 500 hours before failure? than 2.0 days?
(c) If an assembly has been on test for 400 hours without a fail- (b) What is the probability that the time until the next change
ure, what is the probability of a failure in the next 100 hours? is greater 7.0 days?
(d) If 10 assemblies are tested, what is the probability that (c) What is the time of the next change that is exceeded with
at least one fails in less than 100 hours? Assume that the probability 90%?
assemblies fail independently. (d) What is the probability that the next change occurs in less than
(e) If 10 assemblies are tested, what is the probability that all 10.0 days, given that it has not yet occurred after 3.0 days?
have failed by 800 hours? Assume that the assemblies fail 4-132. The length of stay at a speciic emergency department
independently. in a hospital in Phoenix, Arizona had a mean of 4.6 hours.
4-126. The time between arrivals of small aircraft at a county Assume that the length of stay is exponentially distributed.
airport is exponentially distributed with a mean of one hour. (a) What is the standard deviation of the length of stay?
(a) What is the probability that more than three aircraft arrive (b) What is the probability of a length of stay of more than 10
within an hour? hours?
(b) If 30 separate one-hour intervals are chosen, what is the (c) What length of stay is exceeded by 25% of the visits?
probability that no interval contains more than three arrivals? 4-133. An article in Journal of National Cancer Institute
(c) Determine the length of an interval of time (in hours) such that [“Breast Cancer Screening Policies in Developing Countries:
the probability that no arrivals occur during the interval is 0.10. A Cost-Effectiveness Analysis for India” (2008, Vol.100(18),