Page 125 - Applied statistics and probability for engineers

Section 3-9/Poisson Distribution 103

(a) What is the probability that there are no surface laws in an 3-172. Orders arrive at a Web site according to a Poisson pro-
auto’s interior? cess with a mean of 12 per hour. Determine the following:
(b) If 10 cars are sold to a rental company, what is the prob- (a) Probability of no orders in ive minutes.
ability that none of the 10 cars has any surface laws? (b) Probability of 3 or more orders in ive minutes.
(c) If 10 cars are sold to a rental company, what is the prob- (c) Length of a time interval such that the probability of no
ability that at most 1 car has any surface laws? orders in an interval of this length is 0.001.
3-168. The number of failures of a testing instrument 3-173. The article “An Association Between Fine Particles and
from contamination particles on the product is a Poisson ran- Asthma Emergency Department Visits for Children in Seattle”
dom variable with a mean of 0.02 failures per hour. [Environmental Health Perspectives June, 1999 107(6)] used
(a) What is the probability that the instrument does not fail in Poisson models for the number of asthma emergency depart-
an 8-hour shift? ment (ED) visits per day. For the zip codes studied, the mean
(b) What is the probability of at least one failure in a 24-hour day? ED visits were 1.8 per day. Determine the following:
3-169. The number of content changes to a Web site follows a (a) Probability of more than ive visits in a day.
Poisson distribution with a mean of 0.25 per day. (b) Probability of fewer than ive visits in a week.
(a) What is the probability of two or more changes in a day? (c) Number of days such that the probability of at least one
(b) What is the probability of no content changes in ive days? visit is 0.99.
(c) What is the probability of two or fewer changes in ive days? (d) Instead of a mean of 1.8 per day, determine the mean visits
3-170. The number of views of a page on a Web site follows a per day such that the probability of more than ive visits in
Poisson distribution with a mean of 1.5 per minute. a day is 0.1.
(a) What is the probability of no views in a minute? 3-174. Inclusions are defects in poured metal caused by con-
(b) What is the probability of two or fewer views in 10 minutes? taminants. The number of (large) inclusions in cast iron follows
(c) Does the answer to the previous part depend on whether the a Poisson distribution with a mean of 2.5 per cubic millimeter.
10-minute period is an uninterrupted interval? Explain. Determine the following:
3-171. Cabs pass your workplace according to a Poisson pro- (a) Probability of at least one inclusion in a cubic millimeter.
cess with a mean of ive cabs per hour. Suppose that you exit (b) Probability of at least ive inclusions in 5.0 cubic millimeters.
the workplace at 6:00 p.m. Determine the following: (c) Volume of material to inspect such that the probability of at
(a) Probability that you wait more than 10 minutes for a cab. least one inclusion is 0.99.
(b) Probability that you wait fewer than 20 minutes for a cab. (d) Instead of a mean of 2.5 per cubic millimeters, the mean
(c) Mean number of cabs per hour so that the probability that inclusions per cubic millimeter such that the probability of
you wait more than 10 minutes is 0.1. at least one inclusion is 0.95.

Supplemental Exercises

Problem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion
3-175. Let the random variable X be equally likely to (a) What is the distribution of cracked eggs per dozen? Include
assume any of the values 1 8/ , 1 4/ , or 3 8/ . Determine the mean parameter values.
and variance of X. (b) What is the probability that a carton of a dozen eggs results
3-176. Let X denote the number of bits received in error in a complaint?
in a digital communication channel, and assume that X is a (c) What are the mean and standard deviation of the number of
binomial random variable with p = 0 001. . If 1000 bits are cracked eggs in a carton of a dozen eggs?
transmitted, determine the following: 3-179. A total of 12 cells are replicated. Freshly synthe-
(
(a) P X = ) 1 (b) P X( ≥ 1 ) sized DNA cannot be replicated again until mitosis is com-
( pleted. Two control mechanisms have been identiied—one
(c) P X ≤ ) 2 (d) mean and variance of X
positive and one negative—that are used with equal probabil-
3-177. Batches that consist of 50 coil springs from a produc-
ity. Assume that each cell independently uses a control mecha-
tion process are checked for conformance to customer require-
nism. Determine the following probabilities.
ments. The mean number of nonconforming coil springs in
(a) All cells use a positive control mechanism.
a batch is ive. Assume that the number of nonconforming
(b) Exactly half the cells use a positive control mechanism.
springs in a batch, denoted as X, is a binomial random variable.
(c) More than four but fewer than seven cells use a positive
(a) What are n and p?
control mechanism.
(b) What is P X( ≤ 2 )?
(c) What is P X( ≥ 49 )? 3-180. A congested computer network has a 1% chance of
3-178. An automated egg carton loader has a 1% probability losing a data packet, and packet losses are independent events.
of cracking an egg, and a customer will complain if more than An e-mail message requires 100 packets.
one egg per dozen is cracked. Assume that each egg load is an (a) What distribution of data packets must be re-sent? Include
independent event. the parameter values.