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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

a. What is the probability that during a given 1-min period, the first operator receives no requests?
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.
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?
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
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.
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
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
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
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
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?
b. What are the expected value and standard deviation of the number of small aircraft that arrive during a 90-min period?
a. What is the probability that exactly 6 small aircraft arrive during a 1-hour period? At least 6? At least 10?
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.
c. What is the (approximate) probability that no sampled computers have the defect?
b. What is the (approximate) probability that more than 10 sampled computers have the defect?
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?
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
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
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
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.]
b. If the forest covers 85,000 acres, what is the expected number of trees in the forest?
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 standard deviation of the number among the 25 calls that involve a fax message?
a. What is the probability that the family has x male children?
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 five. 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 earthquake-prone.Suppose that in one metropolitan area, 30% of all homeowners are insured against earthquake damage. Four homeowners?a. Find the probability distribution of X. [Hint: Let S denote a homeowner who has insurance and F one who does not.Then
b. Draw the corresponding probability histogram.
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
13. A mail-order computer business has six telephone lines. Let X denote the number of lines in use at a specified time.Suppose the pmf of X is as given in the accompanying table.Calculate the probability of each of the following events.a. {at most three lines are in use}b. {fewer than three lines
c. If you are the first person on the standby list (which means you will be the first one to get on the plane if there are any seats available after all ticketed passengers have been accommodated), what is the probability that you will be able to take the flight? What is this probability if you are
b. What is the probability that not all ticketed passengers who show up can be accommodated?
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.
128. Let V denote rainfall volume and W denote runoff volume(both in mm). According to the article “Runoff Quality Analysis of Urban Catchments with Analytical Probability Models” (J. of Water Resource Planning and Management, 2006: 4–14), the runoff volume will be 0 if V d and will be k(V
127. An individual’s credit score is a number calculated based on that person’s credit history which helps a lender determine how much he/she should be loaned or what credit limit should be established for a credit card. An article in the Los Angeles Times gave data which suggested that a beta
126. Let X have a Weibull distribution with parameters 2 and . Show that Y 2X2/2 has a chi-squared distribution with 2. [Hint: The cdf of Y is P(Y y); express this probability in the form P(X g(y)), use the fact that X has cdf of the form in Expression (4.12), and differentiate with
125. A function g(x) is convex if the chord connecting any 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 below.) How does g()
124. Consider an rv X with mean and standard deviation , and let g(X) be a specified function of X. The first-order Taylor series approximation to g(X) in the neighborhood of is
123. 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/)ln(1 U).a. Show that X has an exponential distribution with parameter . [Hint: The cdf of X is F(x) P(X x); X
122. Let X denote the lifetime of a component, with f(x) and F(x) the pdf and cdf of X. The probability that the component fails in the interval (x, x x) is approximately f(x) x. The conditional probability that it fails in (x, x x)given that it has lasted at least x is f(x) x/[1
6. Starting at a fixed 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 five
121. The article “Three Sisters Give Birth on the Same Day”(Chance, Spring 2001, 23–25) used the fact that three Utah sisters had all given birth on March 11, 1998 as a basis for posing some interesting questions regarding birth coincidences.a. Disregarding leap year and assuming that the
120. 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 2.It can
119. In Exercises 111 and 112, as well as many other situations, one has the pdf f(x) of X and wishes to know the pdf of Y h(X). Assume that h() is an invertible function, so that y h(x) can be solved for x to yield x k(y). Then it can be shown that the pdf of Y is g(y) f [k(y)] ⏐k
118.a. Suppose the lifetime X of a component, when measured in hours, has a gamma distribution with parameters and . Let Y the lifetime measured in minutes.Derive the pdf of Y. [Hint: Y y iff X y/60. Use this to obtain the cdf of Y and then differentiate to obtain the pdf.]b. If X has a
117. Let Z have a standard normal distribution and define a new rv Y by Y Z . Show that Y has a normal distribution with parameters and . [Hint: Y y iff Z ? Use this to find the cdf of Y and then differentiate it with respect to y.]
116. The article “Response of SiCf /Si3N4 Composites Under Static and Cyclic Loading—An Experimental and Statistical Analysis” (J. of Engr. Materials and Technology, 1997: 186–193) suggests that tensile strength (MPa) of composites under specified conditions can be modeled by a Weibull
115. Let Ii be the input current to a transistor and I0 be the output current. Then the current gain is proportional to ln(I0/Ii). Suppose the constant of proportionality is 1(which amounts to choosing a particular unit of measurement), so that current gain X ln(I0/Ii). Assume X is normally
114. Suppose a particular state allows individuals filing tax returns to itemize deductions only if the total of all itemized deductions is at least $5000. Let X (in 1000s of dollars) be the total of itemized deductions on a randomly chosen form. Assume that X has the pdf f(x; ) k/x x 5 0
113. 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 parameteri (i 1, 2) and p is the proportion of all customers served by facility 1, then the pdf of X the service time of
7. For each random variable defined 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
112. The article “Error Distribution in Navigation” (J. Institute of 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
111. 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 and ?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
110. A component has lifetime X that is exponentially distributed with parameter .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
109. The article “The Prediction of Corrosion by Statistical Analysis of Corrosion Profiles” (Corrosion Science, 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.F(x; , )
108. The article “Determination of the MTF of Positive Photoresists Using the Monte Carlo Method” (Photographic Sci. and Engr., 1983: 254–260) proposes the exponential distribution with parameter .93 as a model for the distribution of a photon’s free path length (m) under certain
107. Let X denote the temperature at which a certain chemical reaction takes place. Suppose that X has pdf f(x) 1 9(4 x2) 1 x 2 0 otherwisea. Sketch the graph of f(x).b. Determine the cdf and sketch it.c. Is 0 the median temperature at which the reaction takes place? If not, is the median
106. The reaction time (in seconds) to a certain stimulus is a continuous random variable with pdf f(x) 3 2 x 12 1 x 3 0 otherwisea. Obtain the cdf.b. What is the probability that reaction time is at most 2.5 sec? Between 1.5 and 2.5 sec?c. Compute the expected reaction time.d. Compute
c. What interval (a,b) includes the central 90% of all grain sizes (so that 5% are below a and 5% are above b)?
b. What is the probability that grain size is between 50 and 80?
a. What is the probability that grain size exceeds 100?
105. The article “Characterization of Room Temperature Damping in Aluminum-Indium Alloys” (Metallurgical Trans., 1993: 1611–1619) suggests that Al matrix grain size(m) for an alloy consisting of 2% indium could be modeled with a normal distribution with a mean value 96 and standard deviation
104. When circuit boards used in the manufacture of compact disc players are tested, the long-run percentage of defectives is 5%. Suppose that a batch of 250 boards has been received and that the condition of any particular board is independent of that of any other board.a. What is the approximate
103. The article “Computer Assisted Net Weight Control”(Quality Progress, 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
102. The breakdown voltage of a randomly chosen diode of a certain type is known to 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
100. 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 years.d.
99. 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 f(y) 2 14y1 1 y2 0 y 12 0 otherwise Compute the following:a. The cdf of Y, and graph
98. 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 millisec, a reasonable assumption is that X is uniformly distributed on the interval [0, 25].a.
97. The following failure time observations (1000s of hours)resulted from accelerated life testing of 16 integrated circuit chips of a certain type:Use the corresponding percentiles of the exponential distribution with 1 to construct a probability plot. Then explain why the plot assesses the
96. 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 be
95. Use a statistical software package to construct a normal probability plot of the tensile ultimate strength data given in Exercise 13 of Chapter 1, and comment.
94. The accompanying observations are precipitation values during March over a 30-year period in Minneapolis–St. Paul.
93. 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 83 (Chapter 1).
92. The article “The Load-Life Relationship for M50 Bearings with Silicon Nitride Ceramic Balls” (Lubrication Engr., 1984: 153–159) reports the accompanying data on bearing load life (million revs.) for bearings tested at a 6.45 kN load.
91. Construct a normal probability plot for the fatigue-crack propagation data given in Exercise 39 (Chapter 1). Does it appear plausible that propagation life has a normal distribution? Explain.
90. The article “A Probabilistic Model of Fracture in Concrete and Size Effects on Fracture Toughness” (Magazine of 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
89. 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. of Quality Technology, 1992:22–26). Would you feel comfortable estimating population mean
88. 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
87. 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?
86. Stress is applied to a 20-in. steel bar that is clamped in a fixed position at each end. Let Y the distance from the left end at which the bar snaps. Suppose Y/20 has a standard beta distribution with E(Y) 10 and V(Y) 10 70.a. What are the parameters of the relevant standard beta
85. Let X have a standard beta density with parameters and .a. Verify the formula for E(X) given in the section.b. Compute E[(1 X)m]. If X represents the proportion of a substance consisting of a particular ingredient, what is the expected proportion that does not consist of this ingredient?
84. 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 5 and 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
83. What condition on and is necessary for the standard beta pdf to be symmetric?
82. The article “The Statistics of Phytotoxic Air Pollutants”(J. Royal Stat. Soc., 1989: 183–198) suggests the lognormal distribution as a model for SO2 concentration above a certain forest. Suppose the parameter values are 1.9 and .9.a. What are the mean value and standard deviation of
81. 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 5 and .1.a. Compute E(X) and V(X).b. Compute P(X 125).c. Compute P(110 X 125).d. What is the
80.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 79?b. Recalling that z is our notation for the 100(1 ) percentile of the standard normal distribution, write an expression for the 100(1 )
79. Let X the 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. Research National Bureau of Standards, vol. 67D,
d. For 0 p 1, give a general expression for the 100pth percentile of the wave-height distribution.
c. What is the median of the wave-height distribution?
b. What is the probability that wave height exceeds its mean value by more than one standard deviation?
a. What is the probability that wave height is at most .5 m?
78. The article “On Assessing the Accuracy of Offshore Wind Turbine Reliability-Based Design Loads from the Environmental Contour Method” (Intl. J. of Offshore and Polar Engr., 2005: 132–140) proposes the Weibull distribution with a 1.817 and b .863 as a model for 1-hour significant wave
c. What is the probability that lifetime is at least 200?Greater than 200?
b. What is the probability that lifetime is at most 100?
a. What are the mean value and standard deviation of lifetime?

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