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introduction to probability statistics
Probability And Statistics For Engineering And The Sciences 8th Edition Jay L Devore, Roger Ellsbury - Solutions
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
a. For f(x1, x2, x3) as given in Example 5.10, compute the joint marginal density function of X, and X3 alone (by integrating over x).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 of
An ecologist wishes to select a point inside a circular sam- pling 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
Use the result of Exercise 28 to show that when X and Y are independent, Cov(X, Y) = Corr(X, Y) = 0.
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 p?
a. Compute the covariance between X and Y in Exercise 9.b. Compute the correlation coefficient p for this X and Y.
a. Compute the covariance for X and Y in Exercise 22.b. Compute p for X and Y in the same exercise.
Show that if X and Y are independent rv's, then E(XY) = E(X) E(Y). Then apply this in Exercise 25.[Hint: Consider the continuous case with f(x, y) = f(x) fy(y).]
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 (3x 0 x 1 1x(x) = 10 otherwise fr(y)= (2y 0 y 1 10 otherwise What is the expected amount of time that the one who
A surveyor wishes to lay out a square region with each side hav- ing 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
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. p(x, y) y 0 5 10 15
Let X1, X2, and X, be the lifetimes of components 1, 2, and 3 in a three-component system.a. How would you define the conditional pdf of X, given that X =x, and X = x,?b. How would you define the conditional joint pdf of X, and X, given that X =x?
Let X, X, X, X, X, and X denote the numbers of blue, brown, green, orange, red, and yellow M&M candies, respectively, in a sample of size n. Then these X's have a multinomial distribution. According to the M&M Web site, the color proportions are p = .24, p = .13, p = .16, .20, ps .13, and p = .14.
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 num- ber of hoses on the full-service island in
Each of n specimens is to be weighed twice on the same scale. Let X and Y, denote the two observed weights for the ith specimen. Suppose X, and Y, are independent of one another, each normally distributed with mean value , (the true weight of specimen i) and variance .a. Show that the maximum
Let X1, X2, X be a random sample from a pdf f(x) that is symmetric about , so that X is an unbiased estimator of u. If n is large, it can be shown that V(X) = 1/(4n[f()]).a. Compare V() to V(A) when the underlying distribution is normal.b. When the underlying pdf is Cauchy (see Example 6.7). V(X) =
In Chapter 3, we defined a negative binomial rv as the num- ber of failures that occur before the rth success in a sequence of independent and identical success/failure trials. The probability mass function (pmf) of X is nb(x; r; p) = x+r-1 + x 1 ) p (1 - p) x = 0, 1, 2, ... xa. Suppose that r2.
Suppose the true average growth of one type of plant during a 1-year period is identical to that of a second type, but the variance of growth for the first type is o, whereas for the second type the variance is 402.Let X,.... X be m independent growth observations on the first type [so E(X) =,
Use this fact to con- struct an unbiased estimator of 0 based on X (and use rules of expected value to show that it is unbiased).b. Estimate from the following n = 10 observations on vibratory stress of a turbine blade under specified conditions: 16.88 10.23 4.59 6.66 13.68 14.23 19.87 9.40 6.51
Let X, X, ..., X represent a random sample from a Rayleigh distribution with pdf f(x; 0) = === x -x(20) x>0a. It can be shown that E(X) =
Consider a random sample X,..., X, from the pdf f(x; 0) = .5(1 +0x) -1x1 where -101 (this distribution arises in particle physics). Show that = 3X is an unbiased estimator of 0.[Hint: First determine = E(X) = E(X).]
Suppose a certain type of fertilizer has an expected yield per acre of , with variance o, whereas the expected yield for a second type of fertilizer is , with the same variance . Let S and S denote the sample variances of yields based on sample sizes n and n, respectively, of the two fertilizers.
Of n, randomly selected male smokers, X, smoked filter cig- arettes, whereas of n, randomly selected female smokers, X, smoked filter cigarettes. Let p, and p2 denote the probabili- ties that a randomly selected male and female, respectively, smoke filter cigarettes.a. Show that (X,/n)-(X/n) is an
Using a long rod that has length , you are going to lay out a square plot in which the length of each side is u. Thus the area of the plot will be u. However, you do not know the value of , so you decide to make n independent measure- ments X, X...., X, of the length. Assume that each X, has mean
Each of 150 newly manufactured items is examined and the number of scratches per item is recorded (the items are sup- posed to be free of scratches), yielding the following data: Number of scratches per item 0 1 2 3 4 5 6 7 Observed frequency 18 37 42 30 13 7 2 1 Let X = the number of scratches on
In a random sample of 80 components of a certain type, 12 are found to be defective.a. Give a point estimate of the proportion of all such compo- nents that are not defective.b. A system is to be constructed by randomly selecting two of these components and connecting them in series, as shown here.
a. A random sample of 10 houses in a particular area, each of which is heated with natural gas, is selected and the amount of gas (therms) used during the month of January is determined for each house. The resulting observations are 103, 156, 118, 89, 125, 147, 122, 109, 138, 99.Let u denote the
An investigator wishes to estimate the proportion of stu- dents at a certain university who have violated the honor code. Having obtained a random sample of n students, she realizes that asking each, "Have you violated the honor code?" will probably result in some untruthful responses. Consider the
Let X have a Weibull distribution with parameters a and , so E(X) B T(1+1/) V(X) = B{T(1 + 2/a) - [T(1 + 1/a)]}a. Based on a random sample X,..., X, write equations for the method of moments estimators of anda. Show that, once the estimate of a has been obtained, the esti- mate of can be found from
When the population distribution is normal, the statistic median {X-X,....|X-X3/.6745 can be used to estimate . This estimator is more resistant to the effects of outliers (observations far from the bulk of the data) than is the sample standard deviation. Compute both the corresponding point
An estimator is said to be consistent if for any > 0, P(0-0) 0 as n . That is, is consistent if, as the sample size gets larger, it is less and less likely that will be further than from the true value of 0.Show that X is a consistent estimator of when K minimizes the mean squared error of this
At time = 0, 20 identical components are tested. The life- time distribution of each is exponential with parameter A. The experimenter then leaves the test facility unmonitored. On his return 24 hours later, the experimenter immediately terminates the test after noticing that y = 15 of the 20 com-
Consider a random sample X, X2,....X, from the shifted exponential pdf x; A, 0) = {Ae-N Ae-Ax-) x0 otherwise Taking 60 gives the pdf of the exponential distribution considered previously (with positive density to the right of zero). An example of the shifted exponential distribution appeared in
Determinea. The maximum likelihood estimator of 0, and then calcu- late the estimate for the vibratory stress data given in that exercise. Is this estimator the same as the unbiased esti- mator suggested in Exercise 15?b. The mle of the median of the vibratory stress distribu- tion. [Hint: First
Let X1, X2,..., X, represent a random sample from the Rayleigh distribution with density function given in Exercise
Let X,,..., X., be a random sample from a gamma distribu- tion with parameters a and B.a. Derive the equations whose solutions yield the maximum likelihood estimators of a and B. Do you think they can be solved explicitly?b. Show that the mle of = a is = X.
Refer to Exercise 25.Suppose we decide to examine another test spot weld. Let X = shear strength of the weld. Use the given data to obtain the mle of P(X400). [Hint: P(X400) = ((400-)/).]
The shear strength of each of ten test spot welds is deter- mined, yielding the following data (psi): 392 376 401 367 389 362 409 415 358 375a. Assuming that shear strength is normally distributed, estimate the true average shear strength and standard deviation of shear strength using the method of
Two different computer systems are monitored for a total of n weeks. Let X, denote the number of breakdowns of the first system during the ith week, and suppose the X's are independent and drawn from a Poisson distribution with parameter . Similarly, let Y, denote the number of break- downs of the
Let X denote the proportion of allotted time that a randomly selected student spends working on a certain aptitude test. Suppose the pdf of X is f(x; 0) (0+1) = 0 0x1 otherwise where -1
Consider the accompanying observations on stream flow (1000s of acre-feet) recorded at a station in Colorado for the period April 1-August 31 over a 31-year span (from an arti- cle in the 1974 volume of Water Resources Research). 127.96 210.07 203.24 108.91 178.21 285.37 100.85 89.59 185.36 126.94
As an example of a situation in which several different statis- tics could reasonably be used to calculate a point estimate, consider a population of N invoices. Associated with each invoice is its "book value," the recorded amount of that invoice. Let T denote the total book value, a known amount.
The article from which the data in Exercise I was extracted also gave the accompanying strength observations for cylinders: 6.1 5.8 7.8 7.1 7.2 9.2 6.6 8.3 7.0 8.3 7.8 8.1 7.4 8.5 8.9 9.8 9.7 14.1 12.6 11.2 Prior to obtaining data, denote the beam strengths by X,,..., X, and the cylinder strengths
We have seen that if E(X) = E(X) == E(X) = , then E(X++ X) = nu. In some applications, the number of X's under consideration is not a fixed num- bern but instead is an rv N. For example, let N = the number of components that are brought into a repair shop on a particular day, and let X, denote the
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
The article "Stochastic Modeling for Pavement Warranty Cost Estimation" (J. of Constr. Engr. and Mgmnt., 2009: 352-359) proposes the following model for the distribution of Y = time to pavement failure. Let X, be the time to fail- ure due to rutting, and X, be the time to failure due to trans-
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 is f(x, y) = [kxy x0, y 0,20 x + y 30 otherwisea. Draw the region of positive density and determine the value of
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: y p(x, y) 12 15 20 12 .05 .05 .10
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 strength
I have three errands to take care of in the Administration Building. Let X, the time that it takes for the ith errand (i=1, 2, 3), and let X the total time in minutes that I spend walking to and from the building and between each errand. Suppose the X's are independent, and normally dis- tributed,
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 ran- dom, the resulting load from the weight is also random; denote
Consider a random sample of size n from a continuous dis- tribution 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
a. Use the general formula for the variance of a linear com- bination to write an expression for V(aX + Y). Then let a = y/y, and show that p-1. [Hint: Variance is always 0, and Cov(X, Y) x y p.]b. By considering V(aX- Y), conclude that p
Consider 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): .83 .88 .88 1.04 1.09 1.12 1.29 1.31 1.48 1.49 1.59 1.62 1.65 1.71 1.76 1.83 Assume that the
A sample of 20 students who had recently taken elementary statistics yielded the following information on the brand of calculator owned (T = Texas Instruments, H = Hewlett Packard, C = Casio, S = Sharp): TTH TCTTSCH SSTH CTTT HTa. Estimate the true proportion of all such students who own a Texas
The accompanying data on flexural strength (MPa) for con- crete beams of a certain type was introduced in Example 1.2. 5.9 7.2 7.3 6.3. 8.1 6.8 7.0 7.6 6.8 6.5 7.0 6.3 7.9 9.0 8.2 8.7 7.8 9.7 7.4 7.7 9.7 7.8 7.7 11.6 11.3 11.8 10.7a. Calculate a point estimate of the mean value of strength for the
Let X1, X2,..., X, be random variables denoting n inde- pendent bids for an item that is for sale. Suppose each X, is uniformly distributed on the interval [100, 200]. If the seller sells to the highest bidder, how much can he expect to earn on the sale? [Hint: Let Y= max(X1, X2,..., X). First find
Let X and Y be independent standard normal random vari- ables, and define a new rv by U = .6X+.8Y.a. Determine Corr(X, U).b. How would you alter U to obtain Corr(X, U) = p for a specified value of p?
A more accurate approximation to E[h(X, Exercise 93 is ah ++ 12 X)] in Compute this for Y=h(X, X, X, X) given in Exercise 93, and compare it to the leading term h(m).
Let XX, be independent rv's with mean values ...... , and variances ,..., 2.Consider a function h(x,, x,), and use it to define a new rv Y=h(X,..., X). Under rather general conditions on the h function, if the 's are all small relative to the corresponding 's, it can be shown that E(Y) h() and V(Y)
Let A denote the percentage of one constituent in a ran- domly selected rock specimen, and let B denote the per- centage of a second constituent in that same specimen. Suppose D and E are measurement errors in determining the values of A and B so that measured values are X = A + D and Y B+E,
A rock specimen from a particular area is randomly selected and weighed two different times. Let W denote the actual weight and X, and X, the two measured weights. Then X, W + E, and X = W + E, where E and E, are the two measurement errors. Suppose that the Es are independent of one another and of
a. Show that Cov(X, Y + Z) Cov(X, Y) + Cov(X, Z).b. Let X and X, be quantitative and verbal scores on one aptitude exam, and let Y, and Y, be corresponding scores on another exam. If Cov(X, Y) = 5, Cov(X, Y2) = 1, Cov(X, Y) 2, and Cov(X, Y) = 8, what is the covariance between the two total scores X
a. Let X, have a chi-squared distribution with parameter v, (see Section 4.4), and let X, be independent of X, and have a chi-squared distribution with parameter 2.Use the technique of Example 5.21 to show that X + X, has a chi-squared distribution with parameterb. In Exercise 71 of Chapter 4, you
Suppose a randomly chosen individual's verbal score X and quantitative score Y on a nationally administered aptitude examination have a joint pdf (2x+3y) 0x 1,0 y 1 f(x, y) = 0 otherwise You are asked to provide a prediction of the individual's total score X+Y. The error of prediction is the mean
c. Use the fact that V(W) = 0 only if W is a constant to show that p = 1 only if Y = ax + b.
Three different roads feed into a particular freeway entrance. Suppose that during a fixed time period, the num- ber of cars coming from each road onto the freeway is a ran- dom variable, with expected value and standard deviation as given in the table. Road Road 2 Road 3 Expected value 800 1000
A concrete beam may fail either by shear (S) or flexure (F).Suppose that three failed beams are randomly selected and the type of failure is determined for each one. Let of beams among the three selected that failed by shear. List each outcome in the sample space along with the associated value of
Let 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.X 5 the number X 5 the number
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 of cars observed. What are possible X values? List five outcomes and their
For each random variable defined here, describe the set of possible values for the variable, and state whether the variable is discrete.a. of unbroken eggs in a randomly chosen standard egg cartonb. of students on a class list for a particular course who are absent on the first day of classesc. of
An individual named Claudius is located at the point 0 in the accompanying diagram.Using an appropriate randomization device (such as a tetrahedral die, one having four sides), Claudius first moves to one of the four locations B1, B2, B3, B4. Once at one of these locations, another randomization
What are possible values of X?Is X discrete or continuous?b. If moves are allowed also along the diagonal paths connecting 0 to A1, A2, A3, and A4, respectively, answer the questions in part (a).
The number of pumps in use at both a six-pump station and a four-pump station will be determined. Give the possible values for each of the following random variables:a. number of pumps in useb. between the numbers in use at stations 1 and 2c. number of pumps in use at either stationd. of stations
An automobile service facility specializing in engine tune-ups knows that 45% of all tune-ups are done on fourcylinder automobiles, 40% on six-cylinder automobiles, and 15% on eight-cylinder automobiles. Let number of cylinders on the next car to be tuned.a. What is the pmf of X?b. Draw both a line
Airlines sometimes overbook flights. Suppose that for a plane with 50 seats, 55 passengers have tickets. Define the random variable Y as the number of ticketed passengers who actually show up for the flight. The probability mass function of Y appears in the accompanying table.X 5 theb. What is the
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.y 45 46 47 48 49 50 51 52 53 54 55 p(y) .05 .10 .12 .14 .25 .17 .06 .05 .03 .02 .01 x 0 1 2 3 4 5 6 p(x) .10 .15 .20 .25 .20
A contractor is required by a county planning department to submit one, two, three, four, or five forms (depending on the nature of the project) in applying for a building permit. Let of forms required of the next applicant.The probability that y forms are required is known to be proportional to
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 by
Some parts of California are particularly earthquake-prone.Suppose that in one metropolitan area, 25% of all homeowners are insured against earthquake damage. Four homeowners are to be selected at random; let X denote the number among the four who have earthquake insurance.a. Find the probability
There are 15 other outcomes.]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?
A new battery’s voltage may be acceptable (A) or unacceptable (U). A certain flashlight requires two batteries, so batteries will be independently selected and tested until two acceptable ones have been found. Suppose that 90% of all batteries have acceptable voltages. Let Y denote the number of
Two fair six-sided dice are tossed independently. Let of the two tosses (so ,, etc.).a. What is the pmf of M? [Hint: First determine p(1), then p(2), and so on.]b. Determine the cdf of M and graph it.
A library subscribes to two different weekly news magazines, each of which is supposed to arrive in Wednesday’s mail. In actuality, each one may arrive on Wednesday, Thursday, Friday, or Saturday. Suppose the two arrive independently of one another, and for each one ,, , and . Let of days beyond
Three couples and two single individuals have been invited to an investment seminar and have agreed to attend.Suppose the probability that any particular couple or individual arrives late is .4 (a couple will travel together in the same vehicle, so either both people will be on time or else both
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
Your first thought might be that the leading digit X of a randomly selected number would be equally likely to be one of the nine possibilities (a discrete uniform distribution). However, much empirical evidence as well as some theoretical arguments suggest an alternative probability distribution
A consumer organization that evaluates new automobiles customarily reports the number of major defects in each car examined. Let X denote the number of major defects in a randomly selected car of a certain type. The cdf of X is as follows:Calculate the following probabilities directly from the
An insurance company offers its policyholders a number of different premium payment options. For a randomly selected policyholder, let of months between successive payments. The cdf of X is as follows:a. What is the pmf of X?b. Using just the cdf, compute and .
In Example 3.12, let of girls born before the experiment terminates. With and, what is the pmf of Y? [Hint: First list the possible values of Y, starting with the smallest, and proceed until you see a general formula.]
Alvie Singer lives at 0 in the accompanying diagram and has four friends who live at A, B, C, and D. One day Alvie decides to go visiting, so he tosses a fair coin twice to decide which of the four to visit. Once at a friend’s house, he will either return home or else proceed to one of the two
Show that the cdf F(x) is a nondecreasing function; that is, implies that . Under what condition will ?
An appliance dealer sells three different models of upright freezers having 13.5, 15.9, and 19.1 cubic feet of storage space, respectively. Let of storage space purchased by the next customer to buy a freezer. Suppose that X has pmfa. Compute E(X), E(X2 ), and V(X).b. If the price of a freezer
Let X be a Bernoulli rv with pmf as in Example 3.18.a. Compute E(X2).b. Show that .c. Compute E(X79).
A small market orders copies of a certain magazine for its magazine rack each week. Let for the magazine, with pmf Suppose the store owner actually pays $2.00 for each copy of the magazine and the price to customers is $4.00. If magazines left at the end of the week have no salvage value, is it
The n candidates for a job have been ranked 1, 2, 3, . . . , n.Let of a randomly selected candidate, so that X has pmf(this is called the discrete uniform distribution). Compute E(X) and V(X) using the shortcut formula. [Hint: The sum of the first n positive integers is , whereas the sum of their
Let when a fair die is rolled once. If before the die is rolled you are offered either (1/3.5) dollars or dollars, would you accept the guaranteed amount or would you gamble? [Note: It is not generally true that .]
A chemical supply company currently has in stock 100 lb of a certain chemical, which it sells to customers in 5-lb batches. Let of batches ordered by a randomly chosen customer, and suppose that X has pmf Compute E(X) and V(X). Then compute the expected number of pounds left after the next
a. Draw a line graph of the pmf of X in Exercise
Then determine the pmf of and draw its line graph. From these two pictures, what can you say about V(X) and?b. Use the proposition involving to establish a general relationship between V(X) and .
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