Page 124 - Applied statistics and probability for engineers

102 Chapter 3/Discrete Random Variables and Probability Distributions

is also 25 and the standard deviation of the counts is 5 per square centimeter. Consequently,
information on the variability is very easily obtained. Conversely, if the variance of count data
is much greater than the mean of the same data, the Poisson distribution is not a good model
for the distribution of the random variable.

Exercises FOR SECTION 3-9

Problem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion
3-157. Suppose that X has a Poisson distribution with a is the probability that you consume one or more insect frag-
mean of 4. Determine the following probabilities: ments in more than one bar?
(
(a) P X = 0) (b) P X ≤ ) 2 (d) Is the probability of contamination more than twice the
(
(
(c) P X = 4) (d) P X = 8) mean of 14.4 unusual, or can it be considered typical vari-
(
3-158. Suppose that X has a Poisson distribution with a ation? Explain.
3-163. In 1898, L. J. Bortkiewicz published a book enti-
mean of 0.4. Determine the following probabilities:
(
(a) P X = 0) (b) P X ≤ ) 2 tled The Law of Small Numbers. He used data collected over 20
(
years to show that the number of soldiers killed by horse kicks
( (
(c) P X = ) 4 (d) P X = ) 8 each year in each corps in the Prussian cavalry followed a Pois-
3-159. Suppose that the number of customers who enter son distribution with a mean of 0.61.
a bank in an hour is a Poisson random variable, and suppose (a) What is the probability of more than one death in a corps in
(
that P X = ) =0 0 05. Determine the mean and variance of X. a year?
.
3-160. The number of telephone calls that arrive at a phone (b) What is the probability of no deaths in a corps over ive years?
exchange is often modeled as a Poisson random variable. Assume 3-164. The number of laws in bolts of cloth in textile manu-
that on the average there are 10 calls per hour. facturing is assumed to be Poisson distributed with a mean of
(a) What is the probability that there are exactly 5 calls in one 0.1 law per square meter.
hour? (a) What is the probability that there are two laws in one
(b) What is the probability that there are 3 or fewer calls in square meter of cloth?
one hour? (b) What is the probability that there is one law in 10 square
(c) What is the probability that there are exactly 15 calls in two meters of cloth?
hours? (c) What is the probability that there are no laws in 20 square
(d) What is the probability that there are exactly 5 calls in 30 meters of cloth?
minutes? (d) What is the probability that there are at least two laws in
3-161. Astronomers treat the number of stars in a given 10 square meters of cloth?
volume of space as a Poisson random variable. The density in 3-165. When a computer disk manufacturer tests a disk, it
the Milky Way Galaxy in the vicinity of our solar system is one writes to the disk and then tests it using a certiier. The certiier
star per 16 cubic light-years. counts the number of missing pulses or errors. The number of errors
(a) What is the probability of two or more stars in 16 cubic on a test area on a disk has a Poisson distribution with λ = 0 2. .
light-years? (a) What is the expected number of errors per test area?
(b) How many cubic light-years of space must be studied so (b) What percentage of test areas have two or fewer errors?
that the probability of one or more stars exceeds 0.95? 3-166. The number of cracks in a section of interstate highway
3-162. Data from www.centralhudsonlabs.com determined that are signiicant enough to require repair is assumed to follow a
the mean number of insect fragments in 225-gram chocolate Poisson distribution with a mean of two cracks per mile.
bars was 14.4, but three brands had insect contamination more (a) What is the probability that there are no cracks that require
than twice the average. See the U.S. Food and Drug Admin- repair in 5 miles of highway?
istration–Center for Food Safety and Applied Nutrition for (b) What is the probability that at least one crack requires
Defect Action Levels for food products. Assume that the num- repair in 1 2/ mile of highway?
ber of fragments (contaminants) follows a Poisson distribution. (c) If the number of cracks is related to the vehicle load on
(a) If you consume a 225-gram bar from a brand at the mean the highway and some sections of the highway have a
contamination level, what is the probability of no insect heavy load of vehicles whereas other sections carry a light
contaminants? load, what do you think about the assumption of a Poisson
(b) Suppose that you consume a bar that is one-ifth the size distribution for the number of cracks that require repair?
tested (45 grams) from a brand at the mean contamination 3-167. The number of surface laws in plastic panels used
level. What is the probability of no insect contaminants? in the interior of automobiles has a Poisson distribution with a
(c) If you consume seven 28.35-gram (one-ounce) bars this mean of 0.05 law per square foot of plastic panel. Assume that
week from a brand at the mean contamination level, what an automobile interior contains 10 square feet of plastic panel.