Page 127 - Applied statistics and probability for engineers

Section 3-9/Poisson Distribution 105

Two patients, assumed to be independent, are each scored. 3-202. Each main bearing cap in an engine contains 4
(a) What is the probability mass function of the total score? bolts. The bolts are selected at random without replacement
(b) What is the probability mass function of the average score? from a parts bin that contains 30 bolts from one supplier and 70
3-195. In a manufacturing process that laminates several bolts from another.
ceramic layers, 1% of the assemblies are defective. Assume that (a) What is the probability that a main bearing cap contains all
the assemblies is independent. bolts from the same supplier?
(a) What is the mean number of assemblies that need to be (b) What is the probability that exactly 3 bolts are from the
checked to obtain ive defective assemblies? same supplier?
(b) What is the standard deviation of the number of assemblies 3-203. Assume that the number of errors along a magnetic
that need to be checked to obtain ive defective assemblies? recording surface is a Poisson random variable with a mean
(c) Determine the minimum number of assemblies that need to of one error every 10 bits. A sector of data consists of 4096
5
be checked so that the probability that at least one defective eight-bit bytes.
assembly is obtained exceeds 0.95.
(a) What is the probability of more than one error in a sector?
3-196. Consider the circuit in Example 2-35. Assume that (b) What is the mean number of sectors until an error occurs?
devices fail independently. What is the probability of two or
fewer failed devices? 3-204. An installation technician for a specialized com-
3-197. Determine the constant c so that the following func- munication system is dispatched to a city only when three or
tion is a probability mass function: f x ( ) = cx for x = 1 2 3 4, , , . more orders have been placed. Suppose that orders follow a
3-198. A manufacturer of a consumer electronics product Poisson distribution with a mean of 0.25 per week for a city
expects 2% of units to fail during the warranty period. A sample with a population of 100,000, and suppose that your city con-
of 500 independent units is tracked for warranty performance. tains a population of 800,000.
(a) What is the probability that none fails during the warranty (a) What is the probability that a technician is required after a
period? one-week period?
(b) What is the expected number of failures during the war- (b) If you are the irst one in the city to place an order, what is
ranty period? the probability that you have to wait more than two weeks
(c) What is the probability that more than two units fail during from the time you place your order until a technician is
the warranty period? dispatched?
3-199. Messages that arrive at a service center for an 3-205. From 500 customers, a major appliance manufacturer
information systems manufacturer have been classiied on the randomly selects a sample without replacement. The company
basis of the number of keywords (used to help route messages) estimates that 25% of the customers will reply to the survey. If
and the type of message, either e-mail or voice. Also, 70% of this estimate is correct, what is the probability mass function of
the messages arrive via e-mail and the rest are voice. the number of customers that will reply?
(a) Assume that the company samples 5 customers.
Number of keywords 0 1 2 3 4
(b) Assume that the company samples 10 customers.
E-mail 0.1 0.1 0.2 0.4 0.2
3-206. It is suspected that some of the totes containing
Voice 0.3 0.4 0.2 0.1 0 chemicals purchased from a supplier exceed the moisture con-
Determine the probability mass function of the number of key- tent target. Samples from 30 totes are to be tested for mois-
words in a message. ture content. Assume that the totes are independent. Determine
the proportion of totes from the supplier that must exceed the
3-200. The random variable X has the following prob-
moisture content target so that the probability is 0.90 that at
ability distribution:
least 1 tote in the sample of 30 fails the test.
x 2 3 5 8 3-207. Messages arrive to a computer server according to
Probability 0.2 0.4 0.3 0.1 a Poisson distribution with a mean rate of 10 per hour. Deter-
mine the length of an interval of time such that the probability
Determine the following: that no messages arrive during this interval is 0.90.
(
(
(a) P X ≤ ) 3 (b) P X > 2 5) 3-208. Flaws occur in the interior of plastic used for auto-
.
. (
(c) P 2 7 < X < 5 1) (d) E X ( ) (e) V X ( ) mobiles according to a Poisson distribution with a mean of
.
3-201. Determine the probability mass function for the random 0.02 law per panel.
variable with the following cumulative distribution function: (a) If 50 panels are inspected, what is the probability that there
⎛ 0 x < 2 are no laws?
⎜ ≤ (b) What is the expected number of panels that need to be
.
⎜ 0 2 2 x < 5 7.
⎜
.
F x ( ) = 0 5. 5 7 ≤ x < 6 5. inspected before a law is found?
⎜ ≤ (c) If 50 panels are inspected, what is the probability that the
.
.
⎜ 0 8 6 5 x < 8 5. number of panels that have one or more laws is fewer than
⎜ ⎝ 1 8 8 5 ≤ x or equal to 2?
.