Page 116 - Applied Statistics And Probability For Engineers

PQ220 6234F.Ch 03 13/04/2002 03:19 PM Page 94

94 CHAPTER 3 DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

(b) What is the probability that the machine will not fail dur- (b) What is the probability that 10 messages are received in
ing a study that includes 500 participants? (Assume one 1.5 hours?
sample per participant.) (c) What is the probability that less than two messages are
3-105. The number of surface flaws in plastic panels used received in one-half hour?
in the interior of automobiles has a Poisson distribution with 3-112. A Web site is operated by four identical computer
a mean of 0.05 flaw per square foot of plastic panel. Assume servers. Only one is used to operate the site; the others are
an automobile interior contains 10 square feet of plastic spares that can be activated in case the active server fails. The
panel. probability that a request to the Web site generates a failure in
(a) What is the probability that there are no surface flaws in the active server is 0.0001. Assume that each request is an in-
an auto’s interior? dependent trial. What is the mean time until failure of all four
(b) If 10 cars are sold to a rental company, what is the proba- computers?
bility that none of the 10 cars has any surface flaws? 3-113. The number of errors in a textbook follow a Poisson
(c) If 10 cars are sold to a rental company, what is the proba- distribution with a mean of 0.01 error per page. What is the
bility that at most one car has any surface flaws? probability that there are three or less errors in 100 pages?
3-106. The number of failures of a testing instrument from 3-114. The probability that an individual recovers from an
contamination particles on the product is a Poisson random illness in a one-week time period without treatment is 0.1.
variable with a mean of 0.02 failure per hour. Suppose that 20 independent individuals suffering from this
(a) What is the probability that the instrument does not fail in illness are treated with a drug and 4 recover in a one-week
an 8-hour shift? time period. If the drug has no effect, what is the probability
(b) What is the probability of at least one failure in a 24-hour that 4 or more people recover in a one-week time period?
day?
3-115. Patient response to a generic drug to control pain is
Supplemental Exercises scored on a 5-point scale, where a 5 indicates complete relief.
Historically the distribution of scores is
3-107. A shipment of chemicals arrives in 15 totes. Three of
the totes are selected at random, without replacement, for an 1 2 3 4 5
inspection of purity. If two of the totes do not conform to 0.05 0.1 0.2 0.25 0.4
purity requirements, what is the probability that at least one of
the nonconforming totes is selected in the sample? Two patients, assumed to be independent, are each scored.
(a) What is the probability mass function of the total score?
3-108. The probability that your call to a service line is an- (b) What is the probability mass function of the average score?
swered in less than 30 seconds is 0.75. Assume that your calls
are independent. 3-116. In a manufacturing process that laminates several
(a) If you call 10 times, what is the probability that exactly 9 ceramic layers, 1% of the assemblies are defective. Assume
of your calls are answered within 30 seconds? that the assemblies are independent.
(b) If you call 20 times, what is the probability that at least 16 (a) What is the mean number of assemblies that need to be
calls are answered in less than 30 seconds? checked to obtain five defective assemblies?
(c) If you call 20 times, what is the mean number of calls that (b) What is the standard deviation of the number of assemblies
are answered in less than 30 seconds? that need to be checked to obtain five defective assemblies?
3-109. Continuation of Exercise 3-108. 3-117. Continuation of Exercise 3-116. Determine the mini-
(a) What is the probability that you must call four times to mum number of assemblies that need to be checked so that the
obtain the first answer in less than 30 seconds? probability of at least one defective assembly exceeds 0.95.
(b) What is the mean number of calls until you are answered 3-118. Determine the constant c so that the following func-
in less than 30 seconds? tion is a probability mass function: f 1x2 cx for x 1, 2, 3, 4.
3-110. Continuation of Exercise 3-109. 3-119. A manufacturer of a consumer electronics product ex-
(a) What is the probability that you must call six times in pects 2% of units to fail during the warranty period. A sample of
order for two of your calls to be answered in less than 30 500 independent units is tracked for warranty performance.
seconds? (a) What is the probability that none fails during the warranty
(b) What is the mean number of calls to obtain two answers in period?
less than 30 seconds? (b) What is the expected number of failures during the
3-111. The number of messages sent to a computer bulletin warranty period?
board is a Poisson random variable with a mean of 5 messages (c) What is the probability that more than two units fail
per hour. during the warranty period?
(a) What is the probability that 5 messages are received in 3-120. Messages that arrive at a service center for an infor-
1 hour? mation systems manufacturer have been classified on the basis