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modern mathematical statistics with applications
Modern Mathematical Statistics With Applications 1st Edition Jay L Devore - Solutions
94. Suppose the number X of tornadoes observed in a particular region during a 1-year period has a Poisson distribution with l 8.a. Compute P(X 5).b. Compute P(6 X9).c. Compute P(10 X).d. What is the probability that the observed number of tornadoes exceeds the expected number by more than 1
96. Consider writing onto a computer disk and then sending it through a certi er that counts the number of missing pulses. Suppose this number X has a Poisson distribution with parameter l.2. (Suggested in Average Sample Number for Semi-Curtailed Sampling Using the Poisson Distribution, J. Qual.
100. The number of people arriving for treatment at an emergency room can be modeled by a Poisson process with a rate parameter of ve 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
103. The article Reliability-Based Service-Life Assessment of Aging Concrete Structures (J. Struct.Engrg., 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 year.a. How
104. Let X have a Poisson distribution with parameter l.Show that E(X) l directly from the de nition of expected value. (Hint: The rst term in the sum equals 0, and then x can be canceled. Now factor out l and show that what is left sums to 1.)
106. Automobiles arrive at a vehicle equipment inspection station according to a Poisson process with rate a 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
107.a. In a Poisson process, what has to happen in both the time interval (0, t) and the interval (t, tt)so that no events occur in the entire interval (0, tt)? Use this and Assumptions 1—3 to write a relationship between P0(tt) and P0(t).b. Use the result of part (a) to write an expression
108.a. Use derivatives of the moment generating function to obtain the mean and variance for the Poisson distribution.b. As discussed in Section 3.4, obtain the Poisson mean and variance from RX(t) ln[MX(t)]. In terms of effort, how does this method compare with the one in part (a)?
109. Show that the binomial moment generating function converges to the Poisson moment generating function if we let nSqand pS0 in such a way that np approaches a value l 0. [Hint: Use the calculus theorem that was used in showing that the binomial probabilities converge to the Poisson
110. Consider a deck consisting of seven cards, marked 1, 2, . . . , 7. Three of these cards are selected at random.De ne an rv W by W the sum of the resulting numbers, and compute the pmf of W. Then compute m and s2. [Hint: Consider outcomes as unordered, so that (1, 3, 7) and (3, 1, 7) are not
111. After shuf ing a deck of 52 cards, a dealer deals out 5. Let X the number of suits represented in the ve-card hand.a. Show that the pmf of X is x 1 2 3 4 p(x) .002 .146 .588 .264[Hint: p(1) 4P(all are spades), p(2) 6P(only spades and hearts with at least one of each), and p(4) 4P(2
112. The negative binomial rv X was de ned as the number of F s preceding the rth S. Let Y the number of trials necessary to obtain the rth S. In the same manner in which the pmf of X was derived, derive the pmf of Y.
114. A friend recently planned a camping trip. He had two ashlights, one that required a single 6-V battery and another that used two size-D batteries. He had previously packed two 6-V and four size-D batteries in his camper. Suppose the probability that any particular battery works is p and that
116. A manufacturer of ashlight 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
121. The purchaser of a power-generating unit requires c consecutive successful start-ups before the unit will be accepted. Assume that the outcomes of individual start-ups are independent of one another.Let p denote the probability that any particular start-up is successful. The random variable of
122. A plan for an executive travelers club has been developed by an airline on the premise that 10% of its current customers would qualify for membership.a. Assuming the validity of this premise, among 25 randomly selected current customers, what is the probability that between 2 and 6
125. A reservation service employs ve information operators who receive requests for information independently of one another, each according to a Poisson process with rate a 2 per minute.a. What is the probability that during a given 1-min period, the rst operator receives no requests?b. What is
126. Grasshoppers are distributed at random in a large eld according to a Poisson distribution with parameter a 2 per square yard. How large should the radius R of a circular sampling region be taken so that the probability of nding at least one in the region equals .99?
127. A newsstand has ordered ve copies of a certain issue of a photography magazine. Let X the number of individuals who come in to purchase this magazine. If X has a Poisson distribution with parameter l 4, what is the expected number of copies that are sold?
130. The generalized negative binomial pmf is given by Let X, the number of plants of a certain species found in a particular region, have this distribution with p .3 and r 2.5. What is P(X 4)? What is the probability that at least one plant is found?
131. De ne a function p(x; l, m) bya. Show that p(x; l, m) satis es the two conditions necessary for specifying a pmf. [Note: If a rm employs two typists, one of whom makes typographical errors at the rate of l per page and the other at rate m per page and they each do half the rm s typing, then
132. The mode of a discrete random variable X with pmf p(x) is that value x* for which p(x) is largest (the most probable x value).a. Let X Bin(n, p). By considering the ratio b(x 1; n, p)/b(x; n, p), show that b(x; n, p) increases with x as long as x np (1 p).Conclude that the mode x* is
134. If X is a hypergeometric rv, show directly from the de nition that E(X)nM/N(consider only the case nM). [Hint: FactornM/Nout of the sum for E(X), and show that the terms inside the sum are of the form h(y; n 1, M 1, N 1), where y x 1.]
135. Use the fact that to prove Chebyshev s inequality, given in Exercise 43 (Section 3.3).a all x 1x m22p1x2 a x:0xm0ks 1x m22p1x2 P1X j 0arm
136. The simple Poisson process of Section 3.7 is characterized by a constant rate a 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 a(t) # t o(t). It can then be shown that the number
137. Suppose a store sells two different coffee makers of a particular brand, a basic model selling for $30 and a fancy one selling for $50. Let X be the number of people among the next 25 purchasing this brand who choose the fancy one. Then h(X) revenue 50X 30(25 X) 20X 750, a linear
138. Let X be a discrete rv with possible values 0, 1, 2, . . . or some subset of these. The function is called the probability generating function [e.g., h(2) 2xp(x), h(3.7) (3.7)xp(x), etc.].a. Suppose X is the number of children born to a family, and p(0) .2, p(1) .5, and p(2)
139. Three couples and two single individuals have been invited to a dinner party. Assume independence of arrivals to the party, and suppose that the probability of any particular individual or any particular couple arriving late is .4 (the two members of a couple arrive together). Let X the
1. Let X denote the amount of time for which a book on 2-hour reserve at a college library is checked out by a randomly selected student and suppose that X has density function Calculate the following probabilities:a. P(X 1)b. P(.5 X1.5)c. P(1.5 X)
2. Suppose the reaction temperature X (in C) in a certain chemical process has a uniformdistribution with A5 and B 5.a. Compute P(X 0).b. Compute P(2.5 X 2.5).c. Compute P(2 X3).d. For k satisfying 5 k k 4 5, compute P(k X k 4).
3. Suppose the error involved in making a certain measurement is a continuous rv X with pdfa. Sketch the graph of f(x).b. Compute P(X 0).c. Compute P(1 X 1).d. Compute P(X.5 or X.5).
4. Let X denote the vibratory stress (psi) on a wind turbine blade at a particular wind speed in a wind tunnel. The article “Blade Fatigue Life Assessment with Application to VAWTS” (J. Solar Energy Engrg., 1982: 107–111) proposes the Rayleigh distribution, with pdf as a model for the X
5. A college professor never finishes his lecture before the end of the hour and always finishes his lectures within 2 min after the hour. Let X the time that elapses between the end of the hour and the end of the lecture and suppose the pdf of X isa. Find the value of k. [Hint: Total area under
7. The time X (min) for a lab assistant to prepare the equipment for a certain experiment is believed to have a uniformdistribution with A25 and B35.a. Write the pdf of X and sketch its graph.b. What is the probability that preparation time exceeds 33 min?c. What is the probability that
8. Commuting to work requires getting on a bus near home and then transferring to a second bus. If the waiting time (in minutes) at each stop has a uniform distribution with A 0 and B 5, then it can be shown that the total waiting time Y has the pdfa. Sketch a graph of the pdf of Y.b. Verify
9. Consider again the pdf of X time headway given in Example 4.5. What is the probability that time headway isa. At most 6 sec?b. More than 6 sec? At least 6 sec?c. Between 5 and 6 sec?
10. A family of pdf’s that has been used to approximate the distribution of income, city population size, and size of firms is the Pareto family. The family has two parameters, k and u, both 0, and the pdf isa. Sketch the graph of f(x; k, u).b. Verify that the total area under the graph equals
11. The cdf of checkout duration X as described in Exercise 1 is Use this to compute the following:a. P(X 1)b. P(.5 X1)c. P(X.5)d. The median checkout duration [solve .5 ]e. F (x) to obtain the density function f(x)
12. The cdf for X (measurement error) of Exercise 3 isa. Compute P(X 0).b. Compute P(1 X 1).c. Compute P(.5 X).d. Verify that f(x) is as given in Exercise 3 by obtaining F (x).e. Verify that .
13. Example 4.5 introduced the concept of time headway in traffic flow and proposed a particular distribution for X the headway between two randomly selected consecutive cars (sec). Suppose that in a different traffic environment, the distribution of time headway has the forma. Determine the
14. Let X denote the amount of space occupied by an article placed in a 1-ft3 packing container. The pdf of X isa. Graph the pdf. Then obtain the cdf of X and graph it.b. What is P(X.5) [i.e., F(.5)]?c. Using part (a), what is P(.25 X.5)? What is P(.25 X.5)?d. What is the 75th percentile of the
15. Answer parts (a)–(d) of Exercise 14 for the random variable X, lecture time past the hour, given in Exercise 5.
16. Let X be a continuous rv with cdf F1x2 μ0 x 0x 4c 1 ln a 4x b d 0 x 41 x 4f 1x2 e 90x811 x2 0 x 1 0 otherwise[This type of cdf is suggested in the article “Variability in Measured Bedload-Transport Rates” (Water Resources Bull., 1985: 39– 48) as a model for a certain
18. Reconsider the distribution of checkout duration X described in Exercises 1 and 11. Compute the following:a. E(X)b. V(X) and sXc. If the borrower is charged an amount h(X) X2 when checkout duration is X, compute the expected charge E[h(X)].
19. Recall the distribution of time headway used in Example 4.5.a. Obtain the mean value of headway and the standard deviation of headway.b. What is the probability that headway is within 1 standard deviation of the mean value?
21. For the distribution of Exercise 14,a. Compute E(X) and sX.b. What is the probability that X is more than 2 standard deviations from its mean value?
22. Consider the pdf of X grade point average given in Exercise 6.a. Obtain and graph the cdf of X.b. From the graph of f(x), what is ?c. Compute E(X) and V(X).
23. Let X have a uniform distribution on the interval[A, B].a. Obtain an expression for the (100p)th percentile.b. Compute E(X), V(X), and sX.c. For n a positive integer, compute E(Xn).m ~
24. Consider the pdf for total waiting time Y for two buses introduced in Exercise 8.a. Compute and sketch the cdf of Y. [Hint: Consider separately 0 y 5 and 5 y10 in computing F(y). A graph of the pdf should be helpful.]b. Obtain an expression for the (100p)th percentile.(Hint: Consider
25. An ecologist wishes to mark off a circular sampling region having radius 10 m. However, the radius of the resulting region is actually a random variable R with pdf What is the expected area of the resulting circular region?
26. The weekly demand for propane gas (in 1000’s of gallons) from a particular facility is an rv X with pdfa. Compute the cdf of X.b. Obtain an expression for the (100p)th percentile.What is the value of ?c. Compute E(X) and V(X).d. If 1.5 thousand gallons are in stock at the beginning of the
27. If the temperature at which a certain compound melts is a random variable with mean value 120C and standard deviation 2C, what are the mean temperature and standard deviation measured in F?(Hint: F 1.8C 32.)
29. At a Website, the waiting time X (in minutes)between hits has pdf f(x) 4e4x, x 0; f(x) 0 otherwise. Find MX(t) and use it to obtain E(X)and V(X).
30. Suppose that the pdf of X isa. Show that .b. The coefficient of skewness is E[(X m)3]/s3.Show that its value for the given pdf is .566.What would the skewness be for a perfectly symmetric pdf?
31. Let X have a uniform distribution on the interval [A, B], so its pdf is f(x) 1/(B A), A xB, f(x) 0 otherwise. Show that the moment generating function of X is
32. Use Exercise 31 to find the pdf f(x) of X if its moment generating function is Explain why you know that your f(x) is uniquely determined by MX(t).
33. If the pdf of a measurement error X is f(x) , show that
34. In Example 4.5 the pdf of X is given as Find the moment generating function and use it to find the mean and variance.
35. For the mgf of Exercise 34, obtain the mean and variance by differentiating RX(t). Compare the answers with the results of Exercise 34.
36. Let X be uniformly distributed on [0, 1]. Find a linear function Y g(X) such that the interval [0, 1] is transformed into [5, 5]. Use the relationship for linear functions MaX b(t) ebtMX(at) to obtain the mgf of Y from the mgf of X. Compare your answer with the result of Exercise 31, and
37. Suppose the pdf of X is Find the moment generating function and use it to find the mean and variance. Compare with Exercise 34, and explain the similarities and differences.
38. Let X be the random variable of Exercise 34. Let Y X .5 and use the relationship MaXb(t) ebtMX(at) to obtain the mgf of Y from the mgf of Exercise 34. Compare with the result of Exercise 37 and explain.f 1x2 e.15e.15x x 0 0 otherwise
40. In each case, determine the value of the constant c that makes the probability statement correct.a. (c) .9838b. P(0 Zc) .291c. P(c Z) .121d. P(c Zc) .668e. P(c) .016
42. Determine za for the following:a. a .0055b. a .09c. a .663
43. If X is a normal rv with mean 80 and standard deviation 10, compute the following probabilities by standardizing:a. P(X 100)b. P(X 80)c. P(65 X100)d. P(70 X)e. P(85 X95)f. P(10)
45. The article “Reliability of Domestic-Waste Biofilm Reactors” (J. Envir. Engrg., 1995: 785–790) suggests that substrate concentration (mg/cm3) of influent to a reactor is normally distributed with m .30 and s .06.a. What is the probability that the concentration exceeds .25?b. What is
48. Human body temperatures for healthy individuals have approximately a normal distribution with mean 98.25F and standard deviation .75F. (The past accepted value of 98.6 Fahrenheit was obtained by converting the Celsius value of 37, which is correct to the nearest integer.)a. Find the 90th
49. The article “Monte Carlo Simulation—Tool for Better Understanding of LRFD” (J. Struct. Engrg., 1993: 1586 –1599) suggests that yield strength (ksi)for A36 grade steel is normally distributed with m43 and s 4.5.a. What is the probability that yield strength is at most 40? Greater than
51. The temperature reading from a thermocouple placed in a constant-temperature medium is normally distributed with mean m, the actual temperature of the medium, and standard deviation s. What would the value of s have to be to ensure that 95%of all readings are within .1 of m?
52. The distribution of resistance for resistors of a certain type is known to be normal, with 10% of all resistors having a resistance exceeding 10.256 ohms and 5% having a resistance smaller than 9.671 ohms. What are the mean value and standard deviation of the resistance distribution?
54. A machine that produces ball bearings has initially been set so that the true average diameter of the bearings it produces is .500 in. A bearing is acceptable if its diameter is within .004 in. of this target value. Suppose, however, that the setting has changed during the course of production,
55. The Rockwell hardness of a metal is determined by impressing a hardened point into the surface of the metal and then measuring the depth of penetration of the point. Suppose the Rockwell hardness of a particular alloy is normally distributed with mean 70 and standard deviation 3. (Rockwell
57. Suppose Appendix Table A.3 contained (z) only for z 0. Explain how you could still computea. P(1.72 Z.55)b. P(1.72 Z.55)Is it necessary to table (z) for z negative? What property of the standard normal curve justifies your answer?
58. Consider babies born in the “normal” range of 37– 43 weeks of gestational age. Extensive data supports the assumption that for such babies born in the United States, birth weight is normally distributed with mean 3432 g and standard deviation 482 g.[The article “Are Babies Normal?”
60. Chebyshev’s inequality, introduced in Exercise 43(Chapter 3), is valid for continuous as well as discrete distributions. It states that for any number k satisfying k 1, (see Exercise 43 in Section 3.3 for an interpretation and Exercise 135 in Chapter 3 Supplementary Exercises for a proof).
64. Suppose only 70% of all drivers in a certain state regularly wear a seat belt. A random sample of 500 drivers is selected. What is the probability thata. Between 320 and 370 (inclusive) of the drivers in the sample regularly wear a seat belt?b. Fewer than 325 of those in the sample regularly
66.a. Show that if X has a normal distribution with parameters m and s, then YaXb (a linear function of X) also has a normal distribution. What are the parameters of the distribution of Y [i.e., E(Y) and V(Y)]? [Hint:Write the cdf of Y, P(Y y), as an integral involving the pdf of X, and then
67. There is no nice formula for the standard normal cdf(z), but several good approximations have been published in articles. The following is from “Approximations for Hand Calculators Using Small Integer Coefficients” (Math. Comput., 1977:214 –222). For 0 z 5.5, .5 exp ec 183z 3512z
68. The moment generating function can be used to find the mean and variance of the normal distribution.a. Use derivatives of MX(t) to verify that E(X) m and V(X) s2.b. Repeat (a) using RX(t) ln [MX(t)], and compare with part (a) in terms of effort.
69. Evaluate the following:a. (6)b. (5/2)c. F(4; 5) (the incomplete gamma function)d. F(5; 4)e. F(0; 4)
70. Let X have a standard gamma distribution with a 7. Evaluate the following:a. P(X 5)b. P(X 5)c. P(X 8)d. P(3 X8)e. P(3 X 8)f. P(X 4 or X 6)
71. Suppose the time spent by a randomly selected student at a campus computer lab has a gamma distribution with mean 20 minutes and variance 80 minutes2.a. What are the values of a and b?b. What is the probability that a student uses the lab for at most 24 minutes?c. What is the probability that a
75. Extensive experience with fans of a certain type used in diesel engines has suggested that the exponential distribution provides a good model for time until failure. Suppose the mean time until failure is 25,000 hours. What is the probability thata. A randomly selected fan will last at least
76. The special case of the gamma distribution in which a is a positive integer n is called an Erlang distribution.If we replace b by 1/l in Expression (4.7), the Erlang pdf is It can be shown that if the times between successive events are independent, each with an exponential distribution with
77. A system consists of five identical components connected in series as shown:As soon as one component fails, the entire system will fail. Suppose each component has a lifetime that is exponentially distributed with l .01 and that components fail independently of one another.Define events Ai
79.a. The event {X2 y} is equivalent to what event involving X itself?b. If X has a standard normal distribution, use part(a) to write the integral that equals P(X2 y).Then differentiate this with respect to y to obtain the pdf of X2 [the square of a N(0, 1) variable].Finally, show that X2 has a
80.a. Find the mean and variance of the gamma distribution by differentiating the moment generating function MX(t).b. Find the mean and variance of the gamma distribution by differentiating RX(t) ln[MX(t)].
81. Find the mean and variance of the gamma distribution using integration to obtain E(X) and E(X2).[Hint: Express the integrand in terms of a gamma density.]
82. The lifetime X (in hundreds of hours) of a certain type of vacuum tube has aWeibull distribution with parameters a2 and b3. Compute the following:a. E(X) and V(X)b. P(X 6)c. P(1.5 X6)(This Weibull distribution is suggested as a model for time in service in “On the Assessment of Equipment
84. Let X the time (in 101 weeks) from shipment of a defective product until the customer returns the product. Suppose that the minimum return time is g 3.5 and that the excess X 3.5 over the minimum has a Weibull distribution with parameters a 2 and b 1.5 (see the Indust. Qual. Control
85. Let X have a Weibull distribution with the pdf from Expression (4.11). Verify that m b(1 1/a).(Hint: In the integral for E(X), make the change of variable y (x/b)a, so that x by1/a.)
86.a. In Exercise 82, what is the median lifetime of such tubes? [Hint: Use Expression (4.12).]b. In Exercise 84, what is the median return time?c. If X has a Weibull distribution with the cdf from Expression (4.12), obtain a general expression for the (100p)th percentile of the distribution.d. In
88. The authors of a paper from which the data in Exercise 25 of Chapter 1 was extracted suggested that a reasonable probability model for drill lifetime was a lognormal distribution with m 4.5 and s .8.a. What are the mean value and standard deviation of lifetime?b. What is the probability
89. Let Xthe hourly median power (in decibels) of received radio signals transmitted between two cities.The authors of the article “Families of Distributions for Hourly Median Power and Instantaneous Power of Received Radio Signals” (J. Res. Nat. Bureau Standards, vol. 67D, 1963: 753–762)
90.a. Use Equation (4.13) to write a formula for the median of the lognormal distribution. What is the median for the power distribution of Exercise 89?b. Recalling that za is our notation for the 100(1a)percentile of the standard normal distribution, write an expression for the 100(1 a)
91. A theoretical justification based on a certain material failure mechanism underlies the assumption that ductile strength X of a material has a lognormal distribution. Suppose the parameters are m 5 and s .1.a. Compute E(X) and V(X).m ~b. Compute P(X 125).c. Compute P(110 X125).d. What is
92. The article “The Statistics of Phytotoxic Air Pollutants”(J. Roy. Statist Soc., 1989: 183–198) suggests the lognormal distribution as a model for SO2 concentration above a certain forest. Suppose the parameter values are m 1.9 and s .9.a. What are the mean value and standard deviation
93. What condition on a and b is necessary for the standard beta pdf to be symmetric?
94. Suppose the proportion X of surface area in a randomly selected quadrate that is covered by a certain plant has a standard beta distribution with a5 and b 2.a. Compute E(X) and V(X).b. Compute P(X.2).c. Compute P(.2 X.4).d. What is the expected proportion of the sampling region not covered
97. The accompanying normal probability plot was constructed from a sample of 30 readings on tension for mesh screens behind the surface of video display tubes used in computer monitors. Does it appear plausible that the tension distribution is normal?
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