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modern mathematical statistics with applications
Modern Mathematical Statistics With Applications 1st Edition Jay L Devore - Solutions
98. Consider the following ten observations on bearing lifetime (in hours):152.7 172.0 172.5 173.3 193.0 204.7 216.5 234.9 262.6 422.6 Construct a normal probability plot and comment on the plausibility of the normal distribution as a model for bearing lifetime (data from “Modified Moment
99. Construct a normal probability plot for the following sample of observations on coating thickness for low-viscosity paint (“Achieving a Target Value for a Manufacturing Process: A Case Study,”J. Qual. Tech., 1992: 22–26). Would you feel comfortable estimating population mean thickness
100. The article “A Probabilistic Model of Fracture in Concrete and Size Effects on Fracture Toughness”(Mag. Concrete Res., 1996: 311–320) gives arguments for why the distribution of fracture toughness in concrete specimens should have a Weibull distribution and presents several histograms of
101. Construct a normal probability plot for the fatigue crack propagation data given in Exercise 36 of Chapter 1. Does it appear plausible that propagation life has a normal distribution? Explain.
102. The article “The Load-Life Relationship for M50 Bearings with Silicon Nitride Ceramic Balls” (Lubricat.Engrg., 1984: 153–159) reports the accompanying data on bearing load life (million revs.) for bearings tested at a 6.45-kN load.47.1 68.1 68.1 90.8 103.6 106.0 115.0 126.0 146.6 229.0
103. Construct a probability plot that will allow you to assess the plausibility of the lognormal distribution as a model for the rainfall data of Exercise 80 in Chapter 1.
104. The accompanying observations are precipitation values during March over a 30-year period in Minneapolis–St. Paul..77 1.20 3.00 1.62 2.81 2.48 1.74 .47 3.09 1.31 1.87 .96.81 1.43 1.51 .32 1.18 1.89 1.20 3.37 2.10 .59 1.35 .90 1.95 2.20 .52 .81 4.75 2.05a. Construct and interpret a normal
105. Use a statistical software package to construct a normal probability plot of the shower-flow rate data given in Exercise 13 of Chapter 1, and comment.
106. Let the ordered sample observations be denoted by y1, y2, . . . , yn (y1 being the smallest and yn the largest). Our suggested check for normality is to plot the (1[(i .5)/n], yi) pairs. Suppose we believe that the observations come from a distribution with mean 0, and let w1, . . . , wn
107. The following failure time observations (1000’s of hours) resulted from accelerated life testing of 16 integrated circuit chips of a certain type:82.8 11.6 359.5 502.5 307.8 179.7 242.0 26.5 244.8 304.3 379.1 212.6 229.9 558.9 366.7 204.6 Use the corresponding percentiles of the exponential
108. Relative to the winning time, the time X of another runner in a 10-km race has pdf fX(x) 2/x3, x 1.The reciprocal Y 1/X represents the ratio of the time for the winner divided by the time of the other runner. Find the pdf of Y. Explain why Y also represents the speed of the other runner
109. If X has the pdf fX(x) 2x, 0 x 1, find the pdf of Y 1/X. The distribution of Y is a special case of the Pareto distribution (see Exercise 10).
110. Let X have the pdf fX(x) 2/x3, x 1. Find the pdf of .
111. Let X have the chi-squared distribution with 2 degrees of freedom, so , x 0. Find the pdf of . Suppose you choose a point in two dimensions randomly, with the horizontal and vertical coordinates chosen independently from the standard normal distribution. Then X has the distribution of the
112. If X is distributed as N(m, s2), find the pdf of Y eX. The distribution of Y is lognormal, as discussed in Section 4.5.
113. If the side of a square X is random with the pdf fX(x) x/8, 0 x 4, and Y is the area of the square, find the pdf of Y.
114. Let X have the uniform distribution on [0, 1]. Find the pdf of Yln(X).
115. Let X be uniformly distributed on [0, 1]. Find the pdf of Ytan[p(X.5)]. This is called the Cauchy distribution after the famous mathematician.Y 1X fX1x2 12 e1x/22 Y 1X
116. If X is uniformly distributed on [0, 1], find a linear transformation Y cX d such that Y is uniformly distributed on [a, b], where a and b are any two numbers such that a b. Is there another solution? Explain.
117. If X has the pdf fX(x) x/8, 0 x 4, find a transformation Y g(X) such that Y is uniformly distributed on [0, 1].
118. If X is uniformly distributed on [1, 1], find the pdf of .
119. If X is uniformly distributed on [1, 1], find the pdf of Y X2.
120. Ann is expected at 7:00 pm after an all-day drive.She may be as much as one hour early or as much as three hours late. Assuming that her arrival time X is uniformly distributed over that interval, find the pdf of , the unsigned difference between her actual and predicted arrival times.
121. If X is uniformly distributed on [1, 3], find the pdf of Y X2.
122. If X is distributed as N(0, 1), find the pdf of .
123. A circular target has radius 1 foot. Assume that you hit the target (we shall ignore misses) and that the probability of hitting any region of the target is proportional to the region’s area. If you hit the target at a distance Y from the center, then let X pY2 be the corresponding circular
124. In Exercise 123, suppose instead that Y is uniformly distributed on [0, 1]. Find the pdf of X pY2. Geometrically speaking, why should X have a pdf that is unbounded near 0?
125. Let X have the geometric distribution with pmf pX(x) (1 p)xp, x 0, 1, 2, . . . . Find the pmf of Y X 1. The resulting distribution is also referred to as geometric (see Example 3.10).
126. Let X have a binomial distribution with n 1(Bernoulli distribution). That is, X has pmf b(x; 1, p). If Y 2X 1, find the pmf of Y.
127. Let X the time it takes a read/write head to locate a desired record on a computer disk memory device once the head has been positioned over the correct track. If the disks rotate once every 25 msec, a reasonable assumption is that X is uniformly distributed on the interval [0, 25].a.
128. A 12-in. bar that is clamped at both ends is to be subjected to an increasing amount of stress until it snaps. Let Y the distance from the left end at which the break occurs. Suppose Y has pdf Compute the following:a. The cdf of Y, and graph it.b. P(Y 4), P(Y 6), and P(4 Y6).c. E(Y), E(Y2),
129. Let X denote the time to failure (in years) of a certain hydraulic component. Suppose the pdf of X is f(x) 32/(x 4)3 for x 0.a. Verify that f(x) is a legitimate pdf.b. Determine the cdf.c. Use the result of part (b) to calculate the probability that time to failure is between 2 and 5
130. The completion time X for a certain task has cdf F(x) given bya. Obtain the pdf f(x) and sketch its graph.b. Compute P(.5 X2).c. Compute E(X).
131. The breakdown voltage of a randomly chosen diode of a certain type is knownto be normally distributed with mean value 40 V and standard deviation 1.5 V.a. What is the probability that the voltage of a single diode is between 39 and 42?b. What value is such that only 15% of all diodes have
132. The article “Computer Assisted Net Weight Control”(Qual. Prog., 1983: 22–25) suggests a normal distribution with mean 137.2 oz and standard deviation 1.6 oz, for the actual contents of jars of a certain type. The stated contents was 135 oz.a. What is the probability that a single jar
134. The article “Characterization of Room Temperature Damping in Aluminum-Indium Alloys” (Metallurgical Trans., 1993: 1611–1619) suggests that Al matrix grain size (mm) for an alloy consisting of 2% indium could be modeled with a normal distribution with a mean value 96 and standard
136. Let X denote the temperature at which a certain chemical reaction takes place. Suppose that X has pdfa. Sketch the graph of f(x).b. Determine the cdf and sketch it.f 1x2 •1 914 x2 2 1 x2 0 otherwise f 1x2 •3 2 # 1 x2 1 x3 0 otherwisec. Is 0 the median temperature at which the
138. The article “The Prediction of Corrosion by Statistical Analysis of Corrosion Profiles” (Corrosion Sci., 1985: 305–315) suggests the following cdf for the depth X of the deepest pit in an experiment involving the exposure of carbon manganese steel to acidified seawater.The authors
139. A component has lifetime X that is exponentially distributed with parameter l.a. If the cost of operation per unit time isc, what is the expected cost of operating this component over its lifetime?b. Instead of a constant cost rate c as in part (a), suppose the cost rate is c(1 .5eax) with a
140. The mode of a continuous distribution is the value x* that maximizes f(x).a. What is the mode of a normal distribution with parameters m and s?b. Does the uniform distribution with parameters A and B have a single mode? Why or why not?c. What is the mode of an exponential distribution with
141. The article “Error Distribution in Navigation”(J. Institut. Navigation, 1971: 429– 442) suggests that the frequency distribution of positive errors(magnitudes of errors) is well approximated by an exponential distribution. Let X the lateral position error (nautical miles), which can be
142. In some systems, a customer is allocated to one of two service facilities. If the service time for a customer served by facility i has an exponential distribution with parameter li (i 1, 2) and p is the proportion of all customers served by facility 1, f 1x2 1.1 2e.20x0 q x q then
144. Let Ii be the input current to a transistor and Io be the output current. Then the current gain is proportional to ln(Io/Ii). Suppose the constant of proportionality is 1 (which amounts to choosing a particular unit of measurement), so that current gain X ln(Io/Ii). Assume X is normally
145. The article “Response of SiCf/Si3N4 Composites Under Static and Cyclic Loading—An Experimental and Statistical Analysis” (J. Engrg. Materials Tech., 1997: 186 –193) suggests that tensile strength (MPa) of composites under specified conditions can be modeled by a Weibull distribution
146.a. Suppose the lifetime X of a component, when measured in hours, has a gamma distribution with parameters a andb. Let Y lifetime measured in minutes. Derive the pdf of Y.b. If X has a gamma distribution with parameters a andb, what is the probability distribution of Y cX?
147. Based on data from a dart-throwing experiment, the article “Shooting Darts” (Chance, Summer 1997:16 –19) proposed that the horizontal and vertical errors from aiming at a point target should be independent of one another, each with a normal distribution having mean 0 and variance s2. It
150. Let U have a uniform distribution on the interval[0, 1]. Then observed values having this distribution can be obtained from a computer’s random number generator. Let X(1/l)ln(1 U).a. Show that X has an exponential distribution with parameter l.b. How would you use part (a) and a random
151. Consider an rv X with mean m and standard deviation s, and let g(X) be a specified function of X.The first-order Taylor series approximation to g(X)in the neighborhood of m is The right-hand side of this equation is a linear function of X. If the distribution of X is concentrated in an
152. Afunctiong(x) isconvex if the chord connectingany two points on the function’s graph lies above the graph. When g(x) is differentiable, an equivalent condition is that for every x, the tangent line at x lies entirely on or below the graph. (See the figures
153. Let X have a Weibull distribution with parameters a 2 andb. Show that Y 2X2/b2 has a chisquared distribution with n 2.
154. Let X have the pdf f(x) 1/[p(1 x2)] for q x q (a central Cauchy distribution), and show that Y 1/X has the same distribution. Hint: Consider, the cdf of , then obtain its pdf and show it is identical to the pdf of .
155. A store will order q gallons of a certain liquid product to meet demand during a particular time period. This product can be dispensed to customers in any amount desired, so demand during the period is a continuous random variable X with cdf F(x). There is a fixed cost c0 for ordering the
1. A service station has both self-service and fullservice islands. On each island, there is a single regular unleaded pump with two hoses. Let X denote the number of hoses being used on the self-service island at a particular time, and let Y denote the number of hoses on the full-service island in
Y0}, and compute the probability of this event.d. Compute the marginal pmf of X and of Y. Using pX(x), what is P(X 1)?e. Are X and Y independent rv s? Explain.
2. When an automobile is stopped by a roving safety patrol, each tire is checked for tire wear, and each headlight is checked to see whether it is properly aimed. Let X denote the number of headlights that need adjustment, and let Y denote the number of defective tires.a. If X and Y are independent
3. A certain market has both an express checkout line and a superexpress checkout line. Let X1 denote the number of customers in line at the express checkout at a particular time of day, and let X2 denote the number of customers in line at the superexpress checkout at the same time. Suppose the
6. Let X denote the number of Canon digital cameras sold during a particular week by a certain store. The pmf of X is x 0 1 2 3 4 pX(x) .1 .2 .3 .25 .15 Sixty percent of all customers who purchase these cameras also buy an extended warranty. Let Y denote the number of purchasers during this week
7. The joint probability distribution of the number X of cars and the number Y of buses per signal cycle at a proposed left-turn lane is displayed in the accompanying joint probability table.y p(x, y) 0 1 2 0 .025 .015 .010 1 .050 .030 .020 2 .125 .075 .050 x 3 .150 .090 .060 4 .100 .060 .040 5
8. A stockroom currently has 30 components of a certain type, of which 8 were provided by supplier 1, 10 by supplier 2, and 12 by supplier 3. Six of these are to be randomly selected for a particular assembly.Let X the number of supplier 1 s components selected, Y the number of supplier 2 s
9. Each front tire on a particular type of vehicle is supposed to be lled to a pressure of 26 psi. Suppose the actual air pressure in each tire is a random variable X for the right tire and Y for the left tire, with joint pdf f 1x, y2 e K1x2 y2 2 20 x30, 20 y30 0 otherwisea. What is the value
10. Annie and Alvie have agreed to meet between 5:00 p.m. and 6:00 p.m. for dinner at a local healthfood restaurant. Let XAnnie s arrival time and YAlvie s arrival time. Suppose X and Y are independent with each uniformly distributed on the interval[5, 6].a. What is the joint pdf of X and Y ?b.
11. Two different professors have just submitted nal exams for duplication. Let X denote the number of typographical errors on the rst professor s exam and Y denote the number of such errors on the second exam. Suppose X has a Poisson distribution with parameter l, Y has a Poisson distribution with
12. Two components of a minicomputer have the following joint pdf for their useful lifetimes X and Y:a. What is the probability that the lifetime X of the rst component exceeds 3?f 1x, y2 e xex 11y2 x 0 and y 0 0 otherwise am k0 am kb akbmk 1a b2m A 51x, y2: 0x y 0 16 6. 4b. What
15. Consider a system consisting of three components as pictured. The system will continue to function as long as the rst component functions and either component 2 or component 3 functions. Let X1, X2, and X3 denote the lifetimes of components 1, 2, and 3, respectively. Suppose the Xi s are
16.a. For f (x1, x2, x3) as given in Example 5.10, compute the joint marginal density function of X1 and X3 alone (by integrating over x2).b. What is the probability that rocks of types 1 and 3 together make up at most 50% of the sample?[Hint: Use the result of part (a).]c. Compute the marginal pdf
17. An ecologist wishes to select a point inside a circular sampling region according to a uniform distribution(in practice this could be done by rst selecting a direction and then a distance from the center in that direction). Let X the x coordinate of the point selected and Y the y coordinate
18. An instructor has given a short quiz consisting of two parts. For a randomly selected student, let X the number of points earned on the rst part and Ythe number of points earned on the second part.Suppose that the joint pmf of X and Y is given in the accompanying table.y p(x, y) 0 5 10 15 0
23. Annie and Alvie have agreed to meet for lunch between noon (0:00 p.m.) and 1:00 p.m. Denote Annie s arrival time by X, Alvie s by Y, and suppose X and Y are independent with pdf s What is the expected amount of time that the one who arrives rst must wait for the other person?[Hint: h(X, Y)
24. Suppose that X and Y are independent rv s with moment generating functions MX(t) and MY(t), respectively.If Z X Y, show that MZ(t) MX(t) MY(t).(Hint: Use the proposition on the expected value of a product.)
25. Compute the correlation coef cient r for X and Y of Example 5.15 (the covariance has already been computed).
26.a. Compute the covariance for X and Y in Exercise 18.b. Compute r for X and Y in the same exercise.
27.a. Compute the covariance between X and Y in Exercise 9.b. Compute the correlation coef cient r for this X and Y.
28. Reconsider the minicomputer component lifetimes X and Y as described in Exercise 12. Determine E(XY). What can be said about Cov(X, Y) and r?
29. Show that when X and Y are independent variables, Cov(X, Y) Corr(X, Y ) 0.0 X Y 0. 4 fY1y2 e 2y 0 y1 0 otherwise fX1x2 e 3x2 0 x1 0 otherwise
30.a. Recalling the de nition of s2 for a single rv X, write a formula that would be appropriate for computing the variance of a function h(X, Y) of two random variables. (Hint: Remember that variance is just a special expected value.)b. Use this formula to compute the variance of the recorded
31.a. Use the rules of expected value to show that Cov(aX b, cY d) ac Cov(X, Y).b. Use part (a) along with the rules of variance and standard deviation to show that Corr(aX b, cY d) Corr(X, Y ) when a and c have the same sign.c. What happens if a and c have opposite signs?
32. Show that if Y aX b (a 0), then Corr(X, Y) 1 or 1. Under what conditions will r 1?
33. Show that if X, Y, and Z are rv s and a and b are constants, then Cov(aX bY, Z) a Cov(X, Z) b Cov(Y, Z)
34. Let ZX be the standardized X, ZX(XmX)/sX, and let ZY be the standardized Y, ZY (Y mY)/sY. Use the results of Exercise 31 to show that Corr(X, Y) Cov(ZX, ZY) E(ZXZY).
35. Let ZX be the standardized X, ZX(XmX)/sX, and let ZY be the standardized Y, ZY (Y mY)/sY.a. Show with the help of the previous exercise that E[(ZY rZX)]2 1 r2.b. Use part (a) to show that 1 r1.c. Use part (a) to show that r 1 implies that Y aXb where a 0, and r1 implies that
36. According to an article in the August 30, 2002, issue of the Chron. Higher Ed., 30% of rst-year college students are liberals, 20% are conservatives, and 50% characterize themselves as middle-of-the-road.Choose two students at random, let X be the number of liberals, and let Y be the number of
37. Teresa and Allison each have arrival times uniformly distributed between 12:00 and 1:00. Their times do not in uence each other. If Y is the rst of the two times and X is the second, on a scale of 0 to 1, then the joint pdf of X and Y is f (x, y) 2 for 0 y x 1.a. Find the marginal
38. In Exercise 37,a. Find the marginal density of Y.b. Find the conditional density of X given Y y.c. Find the conditional mean of X given Y y. Is a linear function of y?d. Find the conditional variance of X given Y y.
39. A photographic supply business accepts orders on each of two different phone lines. On each line the waiting time until the rst call is exponentially distributed with mean 1 minute, and the two times are independent of one another. Let X be the shorter of the two waiting times and let Y be the
40. A class has 10 mathematics majors, 6 computer science majors, and 4 statistics majors. A committee of two is selected at random to work on a problem. Let X be the number of mathematics majors and let Y be the number of computer science majors chosen.a. Find the joint probability mass function
41. A stick is 1 foot long. You break it at a point X (measured from the left end) chosen randomly uniformly V1Y X x2 E1Y X x2 E1Y X x2 E1X Y y2 along its length. Then you break the left part at a point Y chosen randomly uniformly along its length.In other words, X is uniformly distributed
42. This is a continuation of the previous exercise.a. Use fY(y) from Exercise 41(c) to get E(Y) and V(Y).b. Use Exercise 41(a) and the theorem of this section to get E(Y) and V(Y).
43. Refer to Exercise 1 and answer the following questions:a. Given that X 1, determine the conditional pmf of Y that is, , , and.b. Given that two hoses are in use at the self-service island, what is the conditional pmf of the number of hoses in use on the full-service island?c. Use the result
45. Suppose that X is uniformly distributed between 0 and 1. Given X x, Y is uniformly distributed between 0 and x2.a. Determine and then . Is x) a linear function of x?b. Find f (x, y) using fX(x) and .c. Find fY (y).
46. This is a continuation of the previous exercise.a. Use fY (y) from Exercise 45(c) to get E(Y)and V(Y).b. Use Exercise 45(a) and the theorem of this section to get E(Y) and V(Y).
47. David and Peter independently choose at random a number from 1, 2, 3, with each possibility equally likely. Let X be the larger of the two numbers, and let Y be the smaller.a. Find p(x, y).b. Find pX(x), x 1, 2, 3.c. Find .d. Find . Is this a linear function of x?e. Find .
48. In Exercise 47 nda. E(X).b. pY(y).c. E(Y ) using pY(y).d. E(Y ) using .e. E(X) E(Y ). Why should this be 4, intuitively?
49. In Exercise 47 nda. .b. . Is this a linear function of y?c. .
50. For a Calculus I class, the nal exam score Y and the average of the four earlier tests X are bivariate normal with mean m1 73, standard deviation s1 12 and mean m2 70, standard deviation s2 15. The correlation is r .71. Find a.b.c.d. , that is, the probability that the nal exam score
51. Let X and Y, reaction times (sec) to two different stimuli, have a bivariate normal distribution with mean m1 20 and standard deviation s1 2 for X and mean m230 and standard deviation s25 for Y. Assume r .8. Find a.b.c.d. P(Y 46X 25)
52. Consider three ping pong balls numbered 1, 2, and 3. Two balls are randomly selected with replacement.If the sum of the two resulting numbers exceeds 4, two balls are again selected. This process continues until the sum is at most 4. Let X and Y sY0Xx s2 Y0Xx mY0Xx P1Y 90 0 X 802 sY0Xx s2
53. Let X be a random digit (0, 1, 2, . . . , 9 are equally likely) and let Y be a random digit not equal to X.That is, the nine digits other than X are equally likely for Y.a. Find pX(x), , pX,Y(x, y).b. Find a formula for . Is this a linear function of x?
54. In our discussion of the bivariate normal distribution, there is an expression for .a. By reversing the roles of X and Y give a similar formula for .b. Both and are linear functions. Show that the product of the two slopes is r2.
55. This week the number X of claims coming into an insurance of ce has a Poisson distribution with mean 100. The probability that any particular claim relates to automobile insurance is .6, independent of any other claim. If Y is the number of automobile claims, then Y is binomial with X trials,
56. In Exercise 55 show that the distribution of Y is Poisson with mean 60. You will need to recognize the Maclaurin series expansion for the exponential function. Use the knowledge that Y is Poisson with mean 60 to nd E(Y) and V(Y).
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