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Probability And Statistics For Engineering And The Sciences 9th Edition Jay L. Devore - Solutions

In commuting to work, a professor must first get on a bus near her house and then transfer 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
a. Use Equation (4.13) to write a formula for the median of the lognormal distribution. What is the median for the load distribution of Exercise 79?b. Recalling that za is our notation for the 100(1 – α) percentile of the standard normal distribution, write an expression for the 100(1 –
Sales delay is the elapsed time between the manufacture of a product and its sale. According to the article "Warranty Claims Data Analysis Considering Sales Delay" (Quality and Reliability Engr. Intl., 2013: 113-123), it is quite common for investigators to model sales delay using a lognormal
As in the case of the Weibull and Gamma distributions, the lognormal distribution can be modified by the introduction of a third parameter λ such that the pdf is shifted to be positive only for x > λ. The article cited in Exercise 4.39 suggested that a shifted lognormal distribution with shift
What condition on α and β is necessary for the standard beta pdf to be symmetric?
Suppose the proportion X of surface area in a randomly selected quadrat that is covered by a certain plant has a standard beta distribution with α = 5 and β = 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
Let X have a standard beta density with parameters a and b. 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?
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) = 100 / 7. a. What are the parameters of the relevant standard beta
A sample of 15 female collegiate golfers was selected and the clubhead velocity (km/hr) while swinging a driver was determined for each one, resulting in the following data ("Hip Rotational Velocities During the Full Golf Swing," J. of Sports Science and Medicine, 2009: 296-299):The corresponding z
The accompanying sample consisting of n = 20 observations on dielectric breakdown voltage of a piece of epoxy resin appeared in the article "Maximum Likelihood Estimation in the 3Parameter Weibull Distribution (IEEE Trans. on Dielectrics and Elec. Insul., 1996: 43-55). The values of (i -.5) / n for
Based on an analysis of sample data, the article "Pedestrians' Crossing Behaviors and Safety at Unmarked Roadways in China" (Accident Analysis and Prevention, 2011: 1927-1936) proposed the pdf f(x) 5 .15e2.15(x21) when x $ 1 as a model for the distribution of X 5 time (sec) spent at the median
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 fracture toughness in concrete specimens should have a Wei bull distribution and presents several histograms of data that appear well
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.
The article "The LoadLife 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.a. Construct a normal probability plot. Is normality plausible? b.
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 in Chapter 1.
The accompanying observations are precipitation values during March over a 30-year period in Minneapolis-St. Paul.a. Construct and interpret a normal probability plot for this data set. b. Calculate the square root of each value and then construct a normal probability plot based on this transformed
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.
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 (ɸ-1((i -.5) / n),yi) pairs. Suppose we believe that the observations come from a distribution with mean 0, and let w1,..., wn be the
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 Î» = 1 to construct a probability plot. Then explain why the plot
Let X 5 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.
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 5 the distance from the left end at which the break occurs. Suppose Y has pdfCompute the following: a. The cdf of Y, and graph it. b. P(Y ‰¤ 4), P(Y > 6), and P(4
A service station has both self-service and full-service 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
Annie and Alvie have agreed to meet between 5:00 p.m. and 6:00 p.m. for dinner at a local health-food restaurant. Let X = Annie's arrival time and Y = Alvie'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
Two different professors have just submitted final exams for duplication. Let X denote the number of typographical errors on the first professor's exam and Y denote the number of such errors on the second exam. Suppose X has a Poisson distribution with parameter µ1, Y has a Poisson
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 first component exceeds 3? b. What are the marginal pdf's of X and Y? Are the two lifetimes independent? Explain. c. What is the probability that
You have two lightbulbs for a particular lamp. Let X = the lifetime of the first bulb and Y = the lifetime of the second bulb (both in 1000s of hours). Suppose that X and Y are independent and that each has an exponential distribution with parameter λ = 1. a. What is the joint pdf of X and Y? b.
Suppose that you have ten lightbulbs, that the lifetime of each is independent of all the other lifetimes, and that each lifetime has an exponential distribution with parameter l. a. What is the probability that all ten bulbs fail before time t? b. What is the probability that exactly k of the ten
Consider a system consisting of three components as pictured. The system will continue to function as long as the first 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
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? c. Compute the marginal pdf of X1 alone.
An ecologist wishes to select a point inside a circular sampling region according to a uniform distribution (in practice this could be done by first 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
Refer to Exercise 1 and answer the following questions: a. Given that X = 1, determine the conditional pmf of Y-i.e., pY | X(0 | 1), pY | X(1 | 1), and pY | X(2 | 1). 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
The joint pdf of pressures for right and left front tires is given in Exercise 9. a. Determine the conditional pdf of Y given that X = x and the conditional pdf of X given that Y = y. b. If the pressure in the right tire is found to be 22 psi, what is the probability that the left tire has a
A large but sparsely populated county has two small hospitals, one at the south end of the county and the other at the north end. The south hospital's emergency room has four beds, whereas the north hospital's emergency room has only three beds. Let X denote the number of south beds occupied at a
Let X1, X2, X3, X4, X5, and X6 denote the numbers of blue, brown, green, orange, red, and yellow M&M candies, respectively, in a sample of size n. Then these Xi's have a multinomial distribution. According to the M&M Web site, the color proportions are p1 = .24, p2 = .13, p3 = .16, p4 = .20, p5 =
Let X1, X2, and X3 be the lifetimes of components 1, 2, and 3 in a three-component system. a. How would you define the conditional pdf of X3 given that X1 = x1 and X2 = x2? b. How would you define the conditional joint pdf of X2 and X3 given that X1 = x1?
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 first part and Y = the number of points earned on the second part. Suppose that the joint pmf of X and Y is given in the accompanying table.a. If the score
The difference between the number of customers in line at the express checkout and the number in line at the super-express checkout in Exercise 3 is X1 - X2. Calculate the expected difference.
Six individuals, including A and B, take seats around a circular table in a completely random fashion. Suppose the seats are numbered 1, . . . , 6. Let X = A's seat number and Y = B's seat number. If A sends a written message around the table to B in the direction in which they are closest, how
A surveyor wishes to lay out a square region with each side having length L. However, because of a measurement error, he instead lays out a rectangle in which the north-south sides both have length X and the east-west sides both have length Y. Suppose that X and Y are independent and that each is
Consider a small ferry that can accommodate cars and buses. The toll for cars is $3, and the toll for buses is $10. Let X and Y denote the number of cars and buses, respectively, carried on a single trip. Suppose the joint distribution of X and Y is as given in the table of Exercise 7. Compute the
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'sWhat is the expected amount of time that the one who arrives first must wait for the other person?
Show that if X and Y are independent rv's, then E(XY) = E(X) ∙ E(Y). Then apply this in Exercise 25.
Compute the correlation coefficient ρ for X and Y of Example 5.16 (the covariance has already been computed).
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 joint
a. Compute the covariance for X and Y in Exercise 22.b. Compute Ï for X and Y in the same exercise.In Exercise 22
a. Compute the covariance between X and Y in Exercise 9.In Exercise 9b. Compute the correlation coefficient Ï for this X and Y.
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 ρ?
a. Recalling the definition of σ2 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. b. Use this formula to compute the variance of the recorded score h(X, Y) [= max(X, Y)] in part (b) of Exercise 22.
a. Use the rules of expected value to show that Cov(aX + b, cY + d) 5 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) 5 Corr(X, Y) when a and c have the same sign. c. What happens if a and c have opposite signs?
Show that if Y = aX + b (a ≠ 0), then Corr(X, Y) = + 1 or - 1. Under what conditions will ρ = + 1?
A particular brand of dishwasher soap is sold in three sizes: 25 oz, 40 oz, and 65 oz. Twenty percent of all purchasers select a 25-oz box, 50% select a 40-oz box, and the remaining 30% choose a 65-oz box. Let X1 and X2 denote the package sizes selected by two independently selected purchasers. a.
It is known that 80% of all brand A external hard drives work in a satisfactory manner throughout the warranty period (are "successes"). Suppose that n = 15 drives are randomly selected. Let X = the number of successes in the sample. The statistic X/n is the sample proportion (fraction) of
Return to the situation described in Exercise 3. a. Determine the marginal pmf of X1, and then calculate the expected number of customers in line at the express checkout. b. Determine the marginal pmf of X2. c. By inspection of the probabilities P(X1 = 4), P(X2 = 0), and P(X1 = 4, X2 = 0), are X1
A box contains ten sealed envelopes numbered 1, . . . , 10. The first five contain no money, the next three each contains $5, and there is a $10 bill in each of the last two. A sample of size 3 is selected with replacement (so we have a random sample), and you get the largest amount in any of the
A company maintains three offices in a certain region, each staffed by two employees. Information concerning yearly salaries (1000s of dollars) is as follows:a. Suppose two of these employees are randomly selected from among the six (without replacement). Determine the sampling distribution of the
Suppose the amount of liquid dispensed by a certain machine is uniformly distributed with lower limit A = 8 oz and upper limit B = 10 oz. Describe how you would carry out simulation experiments to compare the sampling distribution of the (sample) fourth spread for sample sizes n = 5, 10, 20, and 30.
Carry out a simulation experiment using a statistical computer package or other software to study the sampling distribution of when the population distribution is Weibull with α = 2 and β = 5, as in Example 5.20. Consider the four sample sizes n = 5, 10, 20, and 30, and in each case use 1000
Carry out a simulation experiment using a statistical computer package or other software to study the sampling distribution of X when the population distribution is lognormal with E(ln(X)) = 3 and V(ln(X)) = 1. Consider the four sample sizes n = 10, 20, 30, and 50, and in each case use 1000
Young's modulus is a quantitative measure of stiffness of an elastic material. Suppose that for aluminum alloy sheets of a particular type, its mean value and standard deviation are 70 GPa and 1.6 GPa, respectively (values given in the article "Influence of Material Properties Variability on
Refer to Exercise 46. Suppose the distribution is normal (the cited article makes that assumption and even includes the corresponding normal density curve). a. Calculate P(69 ≤ ≤ 71) when n = 16. b. How likely is it that the sample mean diameter exceeds 71 when n = 25?
The National Health Statistics Reports dated Oct. 22, 2008, stated that for a sample size of 277 18-year-old American males, the sample mean waist circumference was 86.3 cm. A somewhat complicated method was used to estimate various population percentiles, resulting in the following values:a. Is it
There are 40 students in an elementary statistics class. On the basis of years of experience, the instructor knows that the time needed to grade a randomly chosen first examination paper is a random variable with an expected value of 6 min and a standard deviation of 6 min. a. If grading times are
The number of customers waiting for gift-wrap service at a department store is an rv X with possible values 0, 1, 2, 3, 4 and corresponding probabilities .1, .2, .3, .25, .15. A randomly selected customer will have 1, 2, or 3 packages for wrapping with probabilities .6, .3, and .1, respectively.
Let X denote the courtship time for a randomly selected female-male pair of mating scorpion flies (time from the beginning of interaction until mating). Suppose the mean value of X is 120 min and the standard deviation of X is 110 min (suggested by data in the article "Should I Stay or Should I Go?
The time taken by a randomly selected applicant for a mortgage to fill out a certain form has a normal distribution with mean value 10 min and standard deviation 2 min. If five individuals fill out a form on one day and six on another, what is the probability that the sample average amount of time
The lifetime of a certain type of battery is normally distributed with mean value 10 hours and standard deviation 1 hour. There are four batteries in a package. What lifetime value is such that the total lifetime of all batteries in a package exceeds that value for only 5% of all packages?
Rockwell hardness of pins of a certain type is known to have a mean value of 50 and a standard deviation of 1.2. a. If the distribution is normal, what is the probability that the sample mean hardness for a random sample of 9 pins is at least 51? b. Without assuming population normality, what is
Suppose the sediment density (g/cm) of a randomly selected specimen from a certain region is normally distributed with mean 2.65 and standard deviation .85 (suggested in "Modeling Sediment and Water Column Interactions for Hydrophobic Pollutants," Water Research, 1984: 1169-1174). a. If a random
The number of parking tickets issued in a certain city on any given weekday has a Poisson distribution with parameter µ = 50. a. Calculate the approximate probability that between 35 and 70 tickets are given out on a particular day. b. Calculate the approximate probability that the total number
A binary communication channel transmits a sequence of "bits" (0s and 1s). Suppose that for any particular bit transmitted, there is a 10% chance of a transmission error (a 0 becoming a 1 or a 1 becoming a 0). Assume that bit errors occur independently of one another. a. Consider transmitting 1000
Suppose the distribution of the time X (in hours) spent by students at a certain university on a particular project is gamma with parameters α = 50 and β = 2. Because α is large, it can be shown that X has approximately a normal distribution. Use this fact to compute the approximate probability
A shipping company handles containers in three different sizes: (1) 27 ft3 (3 × 3 × 3), (2) 125 ft3, and (3) 512 ft3. Let Xi (i = 1, 2, 3) denote the number of type i containers shipped during a given week. With µi = E(Xi) and σ2i = V(Xi), suppose that the mean values and standard deviations
Let X denote the number of Canon SLR cameras sold during a particular week by a certain store. The pmf of X is Sixty percent of all customers who purchase these cameras also buy an extended warranty. Let Y denote the number of purchasers during this week who buy an extended warranty.a. What is P(X
Exercise 26 introduced random variables X and Y, the number of cars and buses, respectively, carried by a ferry on a single trip. The joint pmf of X and Y is given in the table in Exercise 7. It is readily verified that X and Y are independent. a. Compute the expected value, variance, and standard
Manufacture of a certain component requires three different machining operations. Machining time for each operation has a normal distribution, and the three times are independent of one another. The mean values are 15, 30, and 20 min, respectively, and the standard deviations are 1, 2, and 1.5 min,
Refer to Exercise 3. a. Calculate the covariance between X1 = the number of customers in the express checkout and X2 = the number of customers in the super express checkout. b. Calculate V(X1 + X2). How does this compare to V(X1) + V(X2)?
Suppose your waiting time for a bus in the morning is uniformly distributed on [0, 8], whereas waiting time in the evening is uniformly distributed on [0, 10] independent of morning waiting time. a. If you take the bus each morning and evening for a week, what is your total expected waiting
Suppose that when the pH of a certain chemical compound is 5.00, the pH measured by a randomly selected beginning chemistry student is a random variable with mean 5.00 and standard deviation .2. A large batch of the compound is subdivided and a sample given to each student in a morning lab and each
One piece of PVC pipe is to be inserted inside another piece. The length of the first piece is normally distributed with mean value 20 in. and standard deviation .5 in. The length of the second piece is a normal rv with mean and standard deviation 15 in. and .4 in., respectively. The amount of
Two airplanes are flying in the same direction in adjacent parallel corridors. At time t = 0, the first airplane is 10 km ahead of the second one. Suppose the speed of the first plane (km/hr) is normally distributed with mean 520 and standard deviation 10 and the second plane's speed is also
Three different roads feed into a particular freeway entrance. Suppose that during a fixed time period, the number of cars coming from each road onto the freeway is a random variable, with expected value and standard deviation as given in the table.a. What is the expected total number of cars
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.a. What is the probability that there is exactly one car and exactly one bus during a cycle? b. What is the
Consider a random sample of size n from a continuous distribution having median 0 so that the probability of any one observation being positive is .5. Disregarding the signs of the observations, rank them from smallest to largest in absolute value, and let W = the sum of the ranks of the
In Exercise 66, the weight of the beam itself contributes to the bending moment. Assume that the beam is of uniform thickness and density so that the resulting load is uniformly distributed on the beam. If the weight of the beam is random, the resulting load from the weight is also random; denote
I have three errands to take care of in the Administration Building. Let Xi = the time that it takes for the ith errand (i = 1, 2, 3), and let X4 = the total time in minutes that I spend walking to and from the building and between each errand. Suppose the Xi's are independent, and normally
Suppose the expected tensile strength of type-A steel is 105 ksi and the standard deviation of tensile strength is 8 ksi. For type-B steel, suppose the expected tensile strength and standard deviation of tensile strength are 100 ksi and 6 ksi, respectively. Let = the sample average tensile
In an area having sandy soil, 50 small trees of a certain type were planted, and another 50 trees were planted in an area having clay soil. Let X = the number of trees planted in sandy soil that survive 1 year and Y = the number of trees planted in clay soil that survive 1 year. If the probability
A restaurant serves three fixed-price dinners costing $12, $15, and $20. For a randomly selected couple dining at this restaurant, let X = the cost of the man's dinner and Y = the cost of the woman's dinner. The joint pmf of X and Y is given in the following table:a. Compute the marginal pmf's of X
In cost estimation, the total cost of a project is the sum of component task costs. Each of these costs is a random variable with a probability distribution. It is customary to obtain information about the total cost distribution by adding together characteristics of the individual component cost
A health-food store stocks two different brands of a certain type of grain. Let X = the amount (lb) of brand A on hand and Y = the amount of brand B on hand. Suppose the joint pdf of X and Y isa. Draw the region of positive density and determine the value of k. b. Are X and Y independent? Answer by
According to the article "Reliability Evaluation of Hard Disk Drive Failures Based on Counting Processes" (Reliability Engr. and System Safety, 2013: 110-118), particles accumulating on a disk drive come from two sources, one external and the other internal. The article proposed a model in which
Suppose that for a certain individual, calorie intake at breakfast is a random variable with expected value 500 and standard deviation 50, calorie intake at lunch is random with expected value 900 and standard deviation 100, and calorie intake at dinner is a random variable with expected value 2000
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
The mean weight of luggage checked by a randomly selected tourist-class passenger flying between two cities on a certain airline is 40 lb, and the standard deviation is 10 lb. The mean and standard deviation for a business class passenger are 30 lb and 6 lb, respectively. a. If there are 12
Suppose the proportion of rural voters in a certain state who favor a particular gubernatorial candidate is .45 and the proportion of suburban and urban voters favoring the candidate is .60. If a sample of 200 rural voters and 300 urban and suburban voters is obtained, what is the approximate
Let µ denote the true pH of a chemical compound. A sequence of n independent sample pH determinations will be made. Suppose each sample pH is a random variable with expected value µ and standard deviation .1. How many determinations are required if we wish the probability that the sample average
If the amount of soft drink that I consume on any given day is independent of consumption on any other day and is normally distributed with µ = 13 oz and σ = 2 and if I currently have two six-packs of 16-oz bottles, what is the probability that I still have some soft drink left at the end of 2
Refer to Exercise 58, and suppose that the Xi's are independent with each one having a normal distribution. What is the probability that the total volume shipped is at most 100,000 ft3?
A student has a class that is supposed to end at 9:00 a.m. and another that is supposed to begin at 9:10 a.m. Suppose the actual ending time of the 9 a.m. class is a normally distributed rv X1 with mean 9:02 and standard deviation 1.5 min and that the starting time of the next class is also a
Garbage trucks entering a particular waste-management facility are weighed prior to offloading their contents. Let X = the total processing time for a randomly selected truck at this facility (waiting, weighing, and offloading). The article "Estimating Waste Transfer Station Delays Using GPS"

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