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modern mathematical statistics with applications
Modern Mathematical Statistics With Applications 1st Edition Jay L Devore - Solutions
64. At a large university, in the never-ending quest for a satisfactory textbook, the Statistics Department has tried a different text during each of the last three quarters. During the fall quarter, 500 students used the text by Professor Mean; during the winter quarter, 300 students used the text
65. A friend who lives in Los Angeles makes frequent consulting trips to Washington, D.C.; 50% of the time she travels on airline #1, 30% of the time on airline #2, and the remaining 20% of the time on airline#3. For airline #1, ights are late into D.C. 30%of the time and late into L.A. 10% of the
66. Reconsider the credit card scenario of Exercise 47(Section 2.4), and show that A and B are dependent rst by using the de nition of independence and then by verifying that the multiplication property does not hold.
68. In Exercise 15, is any Ai independent of any other Ai? Answer using the multiplication property for independent events.
69. If A and B are independent events, show that A and B are also independent. [Hint: First establish a relationship between P(A B), P(B), and P(A B).]
70. Suppose that the proportions of blood phenotypes in a particular population are as follows:A B AB O.42 .10 .04 .44 Assuming that the phenotypes of two randomly selected individuals are independent of one another, what is the probability that both phenotypes are O?What is the probability that
73. A boiler has ve identical relief valves. The probability that any particular valve will open on demand is .95. Assuming independent operation of the valves, calculate P(at least one valve opens) and P(at least one valve fails to open).
75. Consider the system of components connected as in the accompanying picture. Components 1 and 2 are connected in parallel, so that subsystem works iff either 1 or 2 works; since 3 and 4 are connected in series, that subsystem works iff both 3 and 4 work.If components work independently of one
79. A quality control inspector is inspecting newly produced items for faults. The inspector searches an item for faults in a series of independent xations, each of a xed duration. Given that a aw is actually present, let p denote the probability that the aw is detected during any one xation (this
80.a. A lumber company has just taken delivery on a lot of 10,000 2 4 boards. Suppose that 20% of these boards (2000) are actually too green to be used in rst-quality construction. Two boards are selected at random, one after the other. Let A {the rst board is green} and B {the second board is
81. Refer to the assumptions stated in Exercise 75 and answer the question posed there for the system in the accompanying picture. How would the probability change if this were a subsystem connected in parallel to the subsystem pictured in Figure 2.14(a)?
83. Suppose identical tags are placed on both the left ear and the right ear of a fox. The fox is then let loose for a period of time. Consider the two events C1 {left ear tag is lost} and C2 {right ear tag is lost}. Let p P(C1) P(C2), and assume C1 and C2 are independent events. Derive an
84. A small manufacturing company will start operating a night shift. There are 20 machinists employed by the company.a. If a night crew consists of 3 machinists, how many different crews are possible?b. If the machinists are ranked 1, 2, . . . , 20 in order of competence, how many of these crews
85. A factory uses three production lines to manufacture cans of a certain type. The accompanying table gives percentages of nonconforming cans, categorized by type of nonconformance, for each of the three lines during a particular time period.Line 1 Line 2 Line 3 Blemish 15 12 20 Crack 50 44 40
87. One satellite is scheduled to be launched from Cape Canaveral in Florida, and another launching is scheduled for Vandenberg Air Force Base in California.Let A denote the event that the Vandenberg launch goes off on schedule, and let B represent the event that the Cape Canaveral launch goes off
88. A transmitter is sending a message by using a binary code, namely, a sequence of 0 s and 1 s. Each transmitted bit (0 or 1) must pass through three relays to reach the receiver. At each relay, the probability is.20 that the bit sent will be different from the bit received(a reversal). Assume
89. Individual A has a circle of ve close friends (B, C, D, E, and F). A has heard a certain rumor from outside the circle and has invited the ve friends to a party to circulate the rumor. To begin, A selects one of the ve at random and tells the rumor to the chosen individual. That individual then
90. Refer to Exercise 89. If at each stage the person who currently has the rumor does not know who has already heard it and selects the next recipient at random from all ve possible individuals, what is the probability that F has still not heard the rumor after it has been told ten times at the
91. A chemist is interested in determining whether a certain trace impurity is present in a product. An experiment has a probability of .80 of detecting the impurity if it is present. The probability of not detecting the impurity if it is absent is .90. The prior probabilities of the impurity being
93. One percent of all individuals in a certain population are carriers of a particular disease. A diagnostic test for this disease has a 90% detection rate for carriers and a 5% detection rate for noncarriers. Suppose the test is applied independently to two different blood samples from the same
95. A certain company sends 40% of its overnight mail parcels via express mail service E1. Of these parcels, 2% arrive after the guaranteed delivery time (denote the event late delivery by L). If a record of an overnight mailing is randomly selected from the company s le, what is the probability
96. Refer to Exercise 95. Suppose that 50% of the overnight parcels are sent via express mail service E2 and the remaining 10% are sent via E3. Of those sent via E2, only 1% arrive late, whereas 5% of the parcels handled by E3 arrive late.a. What is the probability that a randomly selected parcel
98. Disregarding the possibility of a February 29 birthday, suppose a randomly selected individual is equally likely to have been born on any one of the other 365 days.a. If ten people are randomly selected, what is the probability that all have different birthdays? That at least two have the same
99. One method used to distinguish between granitic(G) and basaltic (B) rocks is to examine a portion of the infrared spectrum of the sun s energy re ected from the rock surface. Let R1, R2, and R3 denote measured spectrum intensities at three different wavelengths; typically, for granite R1
100. In a Little League baseball game, team As pitcher throws a strike 50% of the time and a ball 50% of the time, successive pitches are independent of one another, and the pitcher never hits a batter.Knowing this, team B s manager has instructed the probability thata. The batter walks on the
101. Four graduating seniors, A, B, C, and D, have been scheduled for job interviews at 10 a.m. on Friday, January 13, at Random Sampling, Inc. The personnel manager has scheduled the four for interview rooms 1, 2, 3, and 4, respectively. Unaware of this, the manager s secretary assigns them to the
102. A particular airline has 10 a.m. ights from Chicago to New York, Atlanta, and Los Angeles. Let A denote the event that the New York ight is full and de ne events B and C analogously for the other two ights. Suppose P(A) .6, P(B) .5, P(C) .4 and the three events are independent. What is
103. A personnel manager is to interview four candidates for a job. These are ranked 1, 2, 3, and 4 in order of preference and will be interviewed in random order. However, at the conclusion of each interview, the manager will know only how the current candidate compares to those previously
107. Jurors may be a priori biased for or against the prosecution in a criminal trial. Each juror is questioned by both the prosecution and the defense (the voir dire process), but this may not reveal bias.Even if bias is revealed, the judge may not excuse the juror for cause because of the narrow
108. Allan and Beth currently have $2 and $3, respectively.A fair coin is tossed. If the result of the toss is H, Allan wins $1 from Beth, whereas if the coin toss results in T, then Beth wins $1 from Allan.This process is then repeated, with a coin toss followed by the exchange of $1, until one of
109. Prove that if [in which case we say that A attracts B ], then[ B attracts A ].
110. Suppose a single gene determines whether the coloring of a certain animal is dark or light. The coloring will be dark if the genotype is either AA or Aa and will be light only if the genotype is aa (so A is dominant and a is recessive).Consider two parents with genotypes Aa and AA. The rst
3. Using the experiment in Example 3.3, de ne two more random variables and list the possible values of each.
4. Let X the number of nonzero digits in a randomly selected zip code. What are the possible values of X?Give three possible outcomes and their associated X values.
5. If the sample space S is an in nite set, does this necessarily imply that any rv X de ned from S will have an in nite set of possible values? If yes, say why. If no, give an example.
6. Starting at a xed time, each car entering an intersection is observed to see whether it turns left (L), right (R), or goes straight ahead (A). The experiment terminates as soon as a car is observed to turn left.Let X the number of cars observed. What are possible X values? List ve outcomes and
7. For each random variable de ned here, describe the set of possible values for the variable, and state whether the variable is discrete.a. X the number of unbroken eggs in a randomly chosen standard egg cartonb. Y the number of students on a class list for a particular course who are absent
9. An individual named Claudius is located at the point 0 in the accompanying diagram.Using an appropriate randomization device (such as a tetrahedral die, one having four sides), Claudius rst moves to one of the four locations B1, B2, B3, B4.Once at one of these locations, he uses another
11. An automobile service facility specializing in engine tune-ups knows that 45% of all tune-ups are done on four-cylinder automobiles, 40% on sixcylinder automobiles, and 15% on eight-cylinder automobiles. Let Xthe number of cylinders on the next car to be tuned.a. What is the pmf of X?b. Draw
12. Airlines sometimes overbook ights. Suppose that for a plane with 50 seats, 55 passengers have tickets.De ne the random variable Y as the number of ticketed passengers who actually show up for the ight.The probability mass function of Y appears in the accompanying table.y 45 46 47 48 49 50 51 52
13. A mail-order computer business has six telephone lines. Let X denote the number of lines in use at a speci ed time. Suppose the pmf of X is as given in the accompanying table.x 0 1 2 3 4 5 6 p(x) .10 .15 .20 .25 .20 .06 .04 Calculate the probability of each of the following events.a. {at most
14. A contractor is required by a county planning department to submit one, two, three, four, or ve forms (depending on the nature of the project) in applying for a building permit. Let Y the number of forms required of the next applicant. The probability that y forms are required is known to be
15. Many manufacturers have quality control programs that include inspection of incoming materials for defects. Suppose a computer manufacturer receives computer boards in lots of ve. Two boards are selected from each lot for inspection. We can represent possible outcomes of the selection process
16. Some parts of California are particularly earthquakeprone.Suppose that in one such area, 30% of all homeowners are insured against earthquake damage.Four homeowners are to be selected at random; let X denote the number among the four who have earthquake insurance.a. Find the probability
19. In Example 3.9, suppose there are only four potential blood donors, of whom only one has type Oblood. Compute the pmf of Y.
22. 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:Calculate the following probabilities directly from the
24. In Example 3.10, let Y the number of girls born before the experiment terminates. With p P(B)and 1 p P(G), what is the pmf of Y? (Hint: First list the possible values of Y, starting with the smallest, and proceed until you see a general formula.)
27. 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)?
28. The pmf for X the number of major defects on a randomly selected appliance of a certain type is x 0 1 2 3 4 p(x) .08 .15 .45 .27 .05 Compute the following:a. E(X)b. V(X) directly from the de nitionc. The standard deviation of Xd. V(X) using the shortcut formula
29. 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 y 0 1 2 3 p(y) .60 .25 .10 .05a. Compute E(Y).b. Suppose an individual with Y violations
30. Refer to Exercise 12 and calculate V(Y) and sY.Then determine the probability that Y is within 1 standard deviation of its mean value.
31. 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 pmf x 13.5 15.9 19.1 p(x) .2 .5 .3a. Compute E(X),
32. Let X be a Bernoulli rv with pmf as in Example 3.17.a. Compute E(X2).b. Show that V(X) p(1 p).c. Compute E(X79).
33. Suppose that the number of plants of a particular type found in a rectangular region (called a quadrat by ecologists) in a certain geographic area is an rv X with pmf p1x2 e c/x3 x 1, 2, 3, . . .0 otherwise Is E(X) nite? Justify your answer (this is another distribution that statisticians
34. A small market orders copies of a certain magazine for its magazine rack each week. Let X demand for the magazine, with pmf x 1 2 3 4 5 6 p(x)Suppose the store owner actually pays $1.00 for each copy of the magazine and the price to customers is $2.00. If magazines left at the end of the week
36. The n candidates for a job have been ranked 1, 2, 3, . . . , n. Let X the rank of a randomly selected candidate, so that X has pmf(this is called the discrete uniform distribution).Compute E(X) and V(X) using the shortcut formula.[Hint: The sum of the rst n positive integers is n(n 1)/2,
37. Let X the outcome when a fair die is rolled once.If before the die is rolled you are offered either(1/3.5) dollars or h(X) 1/X dollars, would you accept the guaranteed amount or would you gamble?[Note: It is not generally true that 1/E(X) E(1/X).]
38. A chemical supply company currently has in stock 100 lb of a certain chemical, which it sells to customers in 5-lb containers. Let X the number of containers ordered by a randomly chosen customer, and suppose that X has pmf Compute E(X) and V(X). Then compute the expected number of pounds
40. Use the de nition in Expression (3.13) to prove that V(aX b) a2 # . [Hint: With h(X) aX b, E[h(X)] am b where m E(X).]
41. Suppose E(X) 5 and E[X(X 1)] 27.5. What isa. E(X2)? [Hint: E[X(X 1)] E[X2 X] E(X2) E(X).]b. V(X)?c. The general relationship among the quantities E(X), E[X(X 1)], and V(X)?
42. Write a general rule for E(X c) where c is a constant.What happens when you let c m, the expected value of X?
43. A result called Chebyshev’s inequality states that for any probability distribution of an rv X and any number k that is at least 1, 1/k2. In words, the probability that the value of X lies at least k standard deviations from its mean is at most 1/k2.a. What is the value of the upper bound for
44. For a new car the number of defects X has the distribution given by the accompanying table. Find MX(t) and use it to nd E(X) and V(X).x 0 1 2 3 4 5 6 p(x) .04 .20 .34 .20 .15 .04 .03
45. In ipping a fair coin let X be the number of tosses to get the rst head. Then p(x) .5x for x 1, 2, 3, . . . . Find MX(t) and use it to get E(X) and V(X).
46. GivenMX(t).2.3et.5e3t, ndp(x), E(X), V(X).
47. Using a calculation similar to the one in Example 3.29 show that, if X has the distribution of Example 3.18, then its mgf is Assuming that Y has mgf MY(t) .75et/(1 .25et), determine the probability mass function pY(y) with the help of the uniqueness property.
48. Let X have the moment generating function of Example 3.29 and let YX1. Recall that X is the number of people who need to be checked to get someone who is Rh, so Y is the number of people checked before the rst Rh person is found. Find MY(t) using the second proposition.
49.If then nd E(X) and V(X) by differentiating MX1t 2 e5t2t2 MX1t 2 pet 1 11 p2eta. MX(t)b. RX(t)
50. Prove the result in the second proposition, MaXb (t) ebtMX(at).
51.Let and let Y (X 5)/2. Find MY(t) and use it to nd E(Y) and V(Y).
52. If you toss a fair die with outcome X, for x 1, 2, 3, 4, 5, 6. Find MX(t).
53. If MX(t) 1/(1t2), nd E(X) and V(X) by differentiating MX(t).
54. Prove that the mean and variance are obtainable from RX(t) ln(MX(t)):
55. Show that g(t) tet cannot be a moment generating function.
56.If then nd E(X) and V(X) by differentiatinga. MX(t)b. RX(t)
57. Let X have the following distribution. Show that the skewness is .6.x 0 1 2 3 p(x) .4 .3 .2 .1
58. Compute the following binomial probabilities directly from the formula for b(x; n, p):a. b(3; 8, .6)b. b(5; 8, .6)c. P(3 X5) when n 8 and p .6d. P(1 X) when n 12 and p .1
59. Use Appendix Table A.1 to obtain the following probabilities:a. B(4; 10, .3)b. b(4; 10, .3)c. b(6; 10, .7)d. P(2 X4) when X Bin(10, .3)e. P(2 X) when X Bin(10, .3)f. P(X 1) when X Bin(10, .7)g. P(2 X 6) when X Bin(10, .3)
60. When circuit boards used in the manufacture of compact disc players are tested, the long-run percentage of defectives is 5%. Let X the number of defective boards in a random sample of size n 25, so X Bin(25, .05).a. Determine P(X 2).b. Determine P(X 5).c. Determine P(1 X4).d. What is
61. A company that produces ne crystal knows from experience that 10% of its goblets have cosmetic aws and must be classi ed as seconds.a. Among six randomly selected goblets, how likely is it that only one is a second?b. Among six randomly selected goblets, what is the probability that at least
62. Suppose that only 25% of all drivers come to a complete stop at an intersection having ashing red lights in all directions when no other cars are visible. What is the probability that, of 20 randomly chosen drivers coming to an intersection under these conditions,a. At most 6 will come to a
63. Exercise 29 (Section 3.3) gave the pmf of Y, the number of traf c 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 citation?c.
67. Suppose that 90% of all batteries from a certain supplier have acceptable voltages. A certain type of ashlight requires two type-D batteries, and the ashlight will work only if both its batteries have acceptable voltages. Among ten randomly selected ashlights, what is the probability that at
70. 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
72.a. For xed n, are there values of p (0 p1) 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.]
73.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?
74. 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 y0 to yn1, and show that the sum equals 1.]
76. An airport limousine can accommodate up to four passengers on any one trip. The company will accept a maximum of six reservations for a trip, and a passenger must have a reservation. From previous records, 20% of all those making reservations do not appear for the trip. Answer the following
77. Refer to Chebyshev s inequality given in Exercise 43 (Section 3.3). Calculate for k 2 and k 3 when X Bin(20, .5), and compare to the corresponding upper bounds. Repeat for X Bin(20, .75).
78. At the end of this section we obtained the mean and variance of a binomial rv using the mgf. Obtain the mean and variance instead from RX(t) ln[MX(t)].
79. Obtain the moment generating function of the number of failures n X in a binomial experiment, and use it to determine the expected number of failures and the variance of the number of failures. Are the expected value and variance intuitively consistent with the expressions for E(X) and
80. 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
81. 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
82. An instructor who taught two sections of statistics last term, the rst 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 rst 15 graded projects.a. What is the
86. A second-stage smog alert has been called in a certain area of LosAngeles County in which there are 50 industrial rms. An inspector will visit 10 randomly selected rms to check for violations of regulations.a. If 15 of the rms are actually violating at least one regulation, what is the pmf of
87. 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 ful lled.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
90. Individual A has a red die and B has a green die(both fair). If they each roll until they obtain ve 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)?
91. For the negative binomial distribution use the moment generating function to derivea. The meanb. The variance
92. If X is a negative binomial rv, then YrX is the total number of trials necessary to obtain r S s. Obtain the mgf of Y and then its mean value and variance.Are the mean and variance intuitively consistent with the expressions for E(X) and V(X)? Explain.
93. Let X, the number of aws on the surface of a randomly selected carpet of a particular type, have a Poisson distribution with parameter l 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 X8)e. P(5 X 8)
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