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applied statistics and probability for engineers

Probability And Statistics For Engineering And The Sciences 7th Edition Dave Ellis, Jay L Devore - Solutions

28. Show that the cdf F(x) is a nondecreasing function; that is, x1 x2 implies that F(x1) F(x2). Under what condition will F(x1) F(x2)?
30. An individual who has automobile insurance from a certain company is randomly selected. Let Y be the number of moving violations for which the individual was cited during the last 3 years. The pmf of Y is
32. An appliance dealer sells three different models of upright freezers having 13.5, 15.9, and 19.1 cubic feet of storage space, respectively. Let X the amount of storage space purchased by the next customer to buy a freezer. Suppose that X has pmfa. Compute E(X), E(X2), and V(X).b. If the price
24. An insurance company offers its policyholders a number of different premium payment options. For a randomly selected policyholder, let X the number of months between successive payments. The cdf of X is as follows:
23. A consumer organization that evaluates new automobiles customarily reports the number of major defects in each car examined. Let X denote the number of major defects in a randomly selected car of a certain type. The cdf of X is as follows:Ï 0 x 0Ô .06 0 x 1Ô .19 1 x 2 F(x) Ì .39
c. What is the most likely value for X?
d. What is the probability that at least two of the four selected have earthquake insurance?
21. Suppose that you read through this year’s issues of the New York Times and record each number that appears in a news article—the income of a CEO, the number of cases of wine produced by a winery, the total charitable contribution of a politician during the previous tax year, the age of a
a. What kind of a distribution does X have (name and values of all parameters)?b. Compute P(X 2), P(X 2), and P(X 2).c. Calculate the mean value and standard deviation of X.
100. A manufacturer of flashlight batteries wishes to control the quality of its product by rejecting any lot in which the proportion of batteries having unacceptable voltage appears to be too high. To this end, out of each large lot(10,000 batteries), 25 will be selected and tested. If at least 5
95. After shuffling a deck of 52 cards, a dealer deals out 5. Let X the number of suits represented in the five-card hand.a. Show that the pmf of X is[Hint: p(1) 4P(all are spades), p(2) 6P(only spades and hearts with at least one of each suit), and p(4) 4P(2 spades one of each other
90. Let X have a Poisson distribution with parameter . Show that E(X) directly from the definition of expected value. [Hint: The first term in the sum equals 0, and then x can be canceled. Now factor out and show that what is left sums to 1.]
91. Suppose that trees are distributed in a forest according to a two-dimensional Poisson process with parameter , the expected number of trees per acre, equal to 80.
a. What is the probability that in a certain quarter-acre plot, there will be at most 16 trees?
b. If the forest covers 85,000 acres, what is the expected number of trees in the forest?
92. Automobiles arrive at a vehicle equipment inspection station according to a Poisson process with rate 10 per hour. Suppose that with probability .5 an arriving vehicle will have no equipment violations.a. What is the probability that exactly ten arrive during the hour and all ten have no
c. Suppose you select a point in the forest and construct a circle of radius .1 mile. Let X the number of trees within that circular region. What is the pmf of X? [Hint:1 sq mile 640 acres.]
93.a. In a Poisson process, what has to happen in both the time interval (0, t) and the interval (t, t t) so that no events occur in the entire interval (0, t t)? Use this and Assumptions 1–3 to write a relationship between P0(t t) and P0(t).b. Use the result of part (a) to write an expression
94. Consider a deck consisting of seven cards, marked 1, 2, . . . , 7. Three of these cards are selected at random. Define an rv W by W the sum of the resulting numbers, and compute the pmf of W. Then compute and 2. [Hint: Consider outcomes as unordered, so that (1, 3, 7) and (3, 1, 7) are not
115. Define a function p(x; , ) by p(x; , ) { e e x 0, 1, 2, . . .0 otherwisea. Show that p(x; , ) satisfies the two conditions necessary for specifying a pmf, [Note: If a firm employs two typists, one of whom makes typographical errors at the rate of per page and the other at rate per
119. Use the fact thatall x(x )2 p(x) x:⏐x⏐k(x )2 p(x)to prove Chebyshev’s inequality given in Exercise 44.
120. The simple Poisson process of Section 3.6 is characterized by a constant rate at which events occur per unit time. A generalization of this is to suppose that the probability of exactly one event occurring in the interval [t, t t] is(t) t o(t). It can then be shown that the number of
109. A reservation service employs five information operators who receive requests for information independently of one another, each according to a Poisson process with rate 2 per minute.
a. What is the probability that during a given 1-min period, the first operator receives no requests?
b. What is the probability that during a given 1-min period, exactly four of the five operators receive no requests?
c. Write an expression for the probability that during a given 1-min period, all of the operators receive exactly the same number of requests.
110. Grasshoppers are distributed at random in a large field according to a Poisson distribution with parameter 2 per square yard. How large should the radius R of a circular sampling region be taken so that the probability of finding at least one in the region equals .99?
89. The article “Reliability-Based Service-Life Assessment of Aging Concrete Structures” (J. Structural Engr., 1993:1600–1621) suggests that a Poisson process can be used to represent the occurrence of structural loads over time. Suppose the mean time between occurrences of loads is .5
b. What is the probability that all of the top five pairs end up playing the same direction?
c. If there are 2n pairs, what is the pmf of X the number among the top n pairs who end up playing east–west?What are E(X) and V(X)?
75. Suppose that p P(male birth) .5. A couple wishes to have exactly two female children in their family. They will have children until this condition is fulfilled.
a. What is the probability that the family has x male children?
b. What is the probability that the family has four children?
c. What is the probability that the family has at most four children?
d. How many male children would you expect this family to have? How many children would you expect this family to have?
a. What is the probability that x of the top 10 pairs end up playing east–west?
73. Twenty pairs of individuals playing in a bridge tournament have been seeded 1, . . . , 20. In the first part of the tournament, the 20 are randomly divided into 10 east–west pairs and 10 north–south pairs.
70. An instructor who taught two sections of engineering statistics last term, the first with 20 students and the second with 30, decided to assign a term project. After all projects had been turned in, the instructor randomly ordered them before grading. Consider the first 15 graded projects.
a. What is the probability that exactly 10 of these are from the second section?
b. What is the probability that at least 10 of these are from the second section?
c. What is the probability that at least 10 of these are from the same section?
d. What are the mean value and standard deviation of the number among these 15 that are from the second section?
e. What are the mean value and standard deviation of the number of projects not among these first 15 that are from the second section?
a. What are the expected value and standard deviation of the number of computers in the sample that have the defect?
b. What is the (approximate) probability that more than 10 sampled computers have the defect?
c. What is the (approximate) probability that no sampled computers have the defect?
85. Suppose small aircraft arrive at a certain airport according to a Poisson process with rate 8 per hour, so that the number of arrivals during a time period of t hours is a Poisson rv with parameter 8t.
a. What is the probability that exactly 6 small aircraft arrive during a 1-hour period? At least 6? At least 10?
b. What are the expected value and standard deviation of the number of small aircraft that arrive during a 90-min period?
c. What is the probability that at least 20 small aircraft arrive during a 21 2-hour period? That at most 10 arrive during this period?
86. The number of people arriving for treatment at an emergency room can be modeled by a Poisson process with a rate parameter of five per hour.a. What is the probability that exactly four arrivals occur during a particular hour?b. What is the probability that at least four people arrive during a
84. Suppose that only .10% of all computers of a certain type experience CPU failure during the warranty period. Consider a sample of 10,000 computers.
78. Individual A has a red die and B has a green die (both fair).If they each roll until they obtain five “doubles” (1–1, . . . , 6–6), what is the pmf of X the total number of times a die is rolled? What are E(X) and V(X)?
79. Let X, the number of flaws on the surface of a randomly selected boiler of a certain type, have a Poisson distribution with parameter 5. Use Appendix Table A.2 to compute the following probabilities:a. P(X 8)b. P(X 8)c. P(9 X)d. P(5 X 8)e. P(5 X 8)
80. Suppose the number X of tornadoes observed in a particular region during a 1-year period has a Poisson distribution with 8.a. Compute P(X 5).b. Compute P(6 X 9).c. Compute P(10 X).d. What is the probability that the observed number of tornadoes exceeds the expected number by more
82. Consider writing onto a computer disk and then sending it through a certifier that counts the number of missing pulses.Suppose this number X has a Poisson distribution with parameter .2. (Suggested in “Average Sample Number for Semi-Curtailed Sampling Using the Poisson Distribution,” J.
a. What is the probability that a disk has exactly one missing pulse?
b. What is the probability that a disk has at least two missing pulses?
c. If two disks are independently selected, what is the probability that neither contains a missing pulse?
69. Each of 12 refrigerators of a certain type has been returned to a distributor because of an audible, high-pitched, oscillating noise when the refrigerator is running. Suppose that 7 of these refrigerators have a defective compressor and the other 5 have less serious problems. If the
75. Suppose that p P(male birth) .5. A couple wishes to have exactly two female children in their family. They will have children until this condition is fulfilled.
c. If there are 2n pairs, what is the pmf of X the number among the top n pairs who end up playing east–west?What are E(X) and V(X)?
b. What is the probability that all of the top five pairs end up playing the same direction?
a. What is the probability that x of the top 10 pairs end up playing east–west?
73. Twenty pairs of individuals playing in a bridge tournament have been seeded 1, . . . , 20. In the first part of the tournament, the 20 are randomly divided into 10 east–west pairs and 10 north–south pairs.
e. What are the mean value and standard deviation of the number of projects not among these first 15 that are from the second section?
d. What are the mean value and standard deviation of the number among these 15 that are from the second section?
b. What is the probability that the family has four children?
c. What is the probability that the family has at most four children?
d. How many male children would you expect this family to have? How many children would you expect this family to have?
c. If two disks are independently selected, what is the probability that neither contains a missing pulse?
b. What is the probability that a disk has at least two missing pulses?
a. What is the probability that a disk has exactly one missing pulse?
82. Consider writing onto a computer disk and then sending it through a certifier that counts the number of missing pulses.Suppose this number X has a Poisson distribution with parameter .2. (Suggested in “Average Sample Number for Semi-Curtailed Sampling Using the Poisson Distribution,” J.
80. Suppose the number X of tornadoes observed in a particular region during a 1-year period has a Poisson distribution with 8.a. Compute P(X 5).b. Compute P(6 X 9).c. Compute P(10 X).d. What is the probability that the observed number of tornadoes exceeds the expected number by more
79. Let X, the number of flaws on the surface of a randomly selected boiler of a certain type, have a Poisson distribution with parameter 5. Use Appendix Table A.2 to compute the following probabilities:a. P(X 8)b. P(X 8)c. P(9 X)d. P(5 X 8)e. P(5 X 8)
78. Individual A has a red die and B has a green die (both fair).If they each roll until they obtain five “doubles” (1–1, . . . , 6–6), what is the pmf of X the total number of times a die is rolled? What are E(X) and V(X)?
c. What is the probability that at least 10 of these are from the same section?
b. What is the probability that at least 10 of these are from the second section?
57. Suppose that 90% of all batteries from a certain supplier have acceptable voltages. A certain type of flashlight requires two type-D batteries, and the flashlight will work only if both its batteries have acceptable voltages. Among ten randomly?
c. What is the probability that at least 2 received a special accommodation?d. What is the probability that the number among the 25 who received a special accommodation is within 2 standard deviations of the number you would expect to be accommodated?e. Suppose that a student who does not receive a
b. What is the probability that at least 1 received a special accommodation?
a. What is the probability that exactly 1 received a special accommodation?
56. The College Board reports that 2% of the 2 million high school students who take the SAT each year receive special accommodations because of documented disabilities (Los Angeles Times, July 16, 2002). Consider a random sample of 25 students who have recently taken the test.
53. Exercise 30 (Section 3.3) gave the pmf of Y, the number of traffic citations for a randomly selected individual insured by a particular company. What is the probability that among 15 randomly chosen such individualsa. At least 10 have no citations?b. Fewer than half have at least one
52. Suppose that 30% of all students who have to buy a text for a particular course want a new copy (the successes!), whereas the other 70% want a used copy. Consider randomly selecting 25 purchasers.a. What are the mean value and standard deviation of the number who want a new copy of the book?b.
60. A toll bridge charges $1.00 for passenger cars and $2.50 for other vehicles. Suppose that during daytime hours, 60% of all vehicles are passenger cars. If 25 vehicles cross the bridge during a particular daytime period, what is the resulting expected toll revenue? [Hint: Let X the number of
a. What is the probability that exactly 10 of these are from the second section?
70. An instructor who taught two sections of engineering statistics last term, the first with 20 students and the second with 30, decided to assign a term project. After all projects had been turned in, the instructor randomly ordered them before grading. Consider the first 15 graded projects.
69. Each of 12 refrigerators of a certain type has been returned to a distributor because of an audible, high-pitched, oscillating noise when the refrigerator is running. Suppose that 7 of these refrigerators have a defective compressor and the other 5 have less serious problems. If the
a. What kind of a distribution does X have (name and values of all parameters)?b. Compute P(X 2), P(X 2), and P(X 2).c. Calculate the mean value and standard deviation of X.
68. A certain type of digital camera comes in either a 3-megapixel version or a 4-megapixel version. A camera store has received a shipment of 15 of these cameras, of which 6 have 3-megapixel resolution. Suppose that 5 of these cameras are randomly selected to be stored behind the counter;the other
67. Refer to Chebyshev’s inequality given in Exercise 44.Calculate P(⏐X ⏐ k) for k 2 and k 3 when X Bin(20, .5), and compare to the corresponding upper bound. Repeat for X Bin(20, .75).
65. Customers at a gas station pay with a credit card (A), debit card (B), or cash (C). Assume that successive customers make independent choices, with P(A) .5, P(B) .2, and P(C) .3.a. Among the next 100 customers, what are the mean and variance of the number who pay with a debit card?Explain
64. Show that E(X) np when X is a binomial random variable.[Hint: First express E(X) as a sum with lower limit x 1.Then factor out np, let y x 1 so that the sum is from y 0 to y n 1, and show that the sum equals 1.]
63.a. Show that b(x; n, 1 p) b(n x; n, p).b. Show that B(x; n, 1 p) 1 B(n x 1; n, p).[Hint: At most x S’s is equivalent to at least (n x) F’s.]c. What do parts (a) and (b) imply about the necessity of including values of p greater than .5 in Appendix Table A.1?
62.a. For fixed n, are there values of p (0 p 1) for which V(X) 0? Explain why this is so.b. For what value of p is V(X) maximized? [Hint: Either graph V(X) as a function of p or else take a derivative.]
c. What is the probability that the number of calls among the 25 that involve a fax transmission exceeds the expected number by more than 2 standard deviations?
84. Suppose that only .10% of all computers of a certain type experience CPU failure during the warranty period. Consider a sample of 10,000 computers.
b. What is the probability that during a given 1-min period, exactly four of the five operators receive no requests?

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