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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

18. Let X1, X2, . . . , Xn be a random sample from a pdf f(x) that is symmetric about , so that is an unbiased estimator of . If n is large, it can be shown that V( ) 1/(4n[ f( )]2).a. Compare V( ) to V(X) when the underlying distribution is normal.b. When the underlying pdf is Cauchy (see
17. In Chapter 3, we defined a negative binomial rv as the number 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)
16. 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 2, whereas for the second type, the variance is 42. Let X1, . . . , Xm be m independent growth observations on the first type [so
15. Let X1, X2, . . . , Xn represent a random sample from a Rayleigh distribution with pdf f(x; ) x ex2/(2) x 0a. It can be shown that E(X2) 2. Use this fact to construct an unbiased estimator of based on X2 i (and use rules of expected value to show that it is unbiased).b. Estimate
13. Consider a random sample X1, . . . , Xn from the pdf f(x; ) .5(1 x) 1 x 1 where 1 1 (this distribution arises in particle physics). Show that ˆ 3X is an unbiased estimator of .[Hint: First determine E(X) E(X).]
c. How would you use the observed values x1 and x2 to estimate the standard error of your estimator?d. If n1 n2 200, x1 127, and x2 176, use the estimator of part (a) to obtain an estimate of p1 p2.e. Use the result of part (c) and the data of part (d) to estimate the standard error of
b. What is the standard error of the estimator in part (a)?
a. Show that (X1/n1) (X2/n2) is an unbiased estimator for p1 p2. [Hint: E(Xi) nipi for i 1, 2.]
11. Of n1 randomly selected male smokers, X1 smoked filter cigarettes, whereas of n2 randomly selected female smokers, X2 smoked filter cigarettes. Let p1 and p2 denote the probabilities that a randomly selected male and female, respectively, smoke filter cigarettes.
10. Using a long rod that has length , you are going to lay out a square plot in which the length of each side is . Thus the area of the plot will be 2. However, you do not know the value of , so you decide to make n independent measurements X1, X2, . . . Xn of the length. Assume that each Xi has
9. Each of 150 newly manufactured items is examined and the number of scratches per item is recorded (the items are supposed to be free of scratches), yielding the following data:
8. 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 components 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
7.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 denote the
6. 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 article 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
5. As an example of a situation in which several different statistics 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
4. The article from which the data of Exercise 1 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 X1, . . . , Xm and the cylinder
3. 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
2. A sample of 20 students who had recently taken elementary statistics yielded the following information on brand of calculator owned (T Texas Instruments, H Hewlett Packard, C Casio, S Sharp):TTHTCTTSCH SSTHCTTTHTa. Estimate the true proportion of all such students who own a Texas
1. The accompanying data on flexural strength (MPa) for concrete 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
38. Each of n specimens is to be weighed twice on the same scale. Let Xi and Yi denote the two observed weights for the ith specimen. Suppose Xi and Yi are independent of one another, each normally distributed with mean value i (the true weight of specimen i) and variance 2.a. Show that the
37. When the sample standard deviation S is based on a random sample from a normal population distribution, it can be shown that E(S) 2/(n1)(n/2)/((n 1)/2)Use this to obtain an unbiased estimator for of the form cS. What is c when n 20?
36. When the population distribution is normal, the statistic median{⏐X1 ⏐, . . . , ⏐Xn ⏐}/.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
35. Let X1, . . . , Xn be a random sample from a pdf that is symmetric about . An estimator for that has been found to perform well for a variety of underlying distributions is the Hodges–Lehmann estimator. To define it, first compute for each i j and each j 1, 2, . . . , n the pairwise
34. The mean squared error of an estimator ˆ is MSE(ˆ) E(ˆ )2. If ˆ is unbiased, then MSE(ˆ) V(ˆ), but in general MSE(ˆ) V(ˆ) (bias)2. Consider the estimatorˆ 2 KS2, where S2 sample variance. What value of K minimizes the mean squared error of this estimator when the
33. At time t 0, there is one individual alive in a certain population. A pure birth process then unfolds as follows. The time until the first birth is exponentially distributed with parameter . After the first birth, there are two individuals alive. The time until the first gives birth again is
32.a. Let X1, . . . , Xn be a random sample from a uniform distribution on [0, ]. Then the mle of is ˆ Y max(Xi).Use the fact that Y y iff each Xi y to derive the cdf of Y. Then show that the pdf of Y max(Xi) is fY(y) {nyn n1 0 y 0 otherwiseb. Use the result of part (a) to
31. An estimator ˆ is said to be consistent if for any ! 0, P(⏐ˆ ⏐ !) 0 0 as n 0 . That is, ˆ is consistent if, as the sample size gets larger, it is less and less likely that ˆ will be further than ! from the true value of . Show that X is a consistent estimator of when 2
30. At time t 0, 20 identical components are put on test. The lifetime distribution of each is exponential with parameter .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
29. Consider a random sample X1, X2, . . . , Xn from the shifted exponential pdf f(x; , ) {e(x) x 0 otherwise Taking 0 gives the pdf of the exponential distribution considered previously (with positive density to the right of zero). An example of the shifted exponential distribution
28. Let X1, X2, . . . , Xn represent a random sample from the Rayleigh distribution with density function given in Exercise 15. Determinea. The maximum likelihood estimator of and then calculate the estimate for the vibratory stress data given in that exercise. Is this estimator the same as the
27. Let X1, . . . , Xn be a random sample from a gamma distribution with parameters and .a. Derive the equations whose solution yields the maximum likelihood estimators of and . Do you think they can be solved explicitly?b. Show that the mle of is ˆ X.
26. 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(X 400).[Hint: P(X 400) ((400 )/).]
25. The shear strength of each of ten test spot welds is determined, 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
24. Refer to Exercise 20. Instead of selecting n 20 helmets to examine, suppose I examine helmets in succession until I have found r 3 flawed ones. If the 20th helmet is the third flawed one (so that the number of helmets examined that were not flawed is x 17), what is the mle of p? Is this
23. Two different computer systems are monitored for a total of n weeks. Let Xi denote the number of breakdowns of the first system during the ith week, and suppose the Xis are independent and drawn from a Poisson distribution with parameter 1. Similarly, let Yi denote the number of breakdowns of
22. 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; ) ( 1)x 0 x 1 0 otherwise where 1 . A random sample of ten students yields data x1 .92, x2 .79, x3 .90, x4 .65, x5 .86, x6
21. Let X have a Weibull distribution with parameters and, so E(X) (1 1/)V(X) 2{(1 2/) [(1 1/)]2}a. Based on a random sample X1, . . . , Xn, write equations for the method of moments estimators of and . Show that, once the estimate of has been obtained, the estimate of
20. A random sample of n bike helmets manufactured by a certain company is selected. Let X the number among the n that are flawed, and let p P(flawed). Assume that only X is observed, rather than the sequence of S’s and F’s.a. Derive the maximum likelihood estimator of p. If n 20 and x
19. An investigator wishes to estimate the proportion of students 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.
18. Let X1, X2, . . . , Xn be a random sample from a pdf f(x) that is symmetric about , so that is an unbiased estimator of . If n is large, it can be shown that V( ) 1/(4n[ f( )]2).a. Compare V( ) to V(X) when the underlying distribution is normal.b. When the underlying pdf is Cauchy (see
17. In Chapter 3, we defined a negative binomial rv as the number 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)
16. 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 2, whereas for the second type, the variance is 42. Let X1, . . . , Xm be m independent growth observations on the first type [so
15. Let X1, X2, . . . , Xn represent a random sample from a Rayleigh distribution with pdf f(x; ) x ex2/(2) x 0a. It can be shown that E(X2) 2. Use this fact to construct an unbiased estimator of based on X2 i (and use rules of expected value to show that it is unbiased).b. Estimate
14. A sample of n captured Pandemonium jet fighters results in serial numbers x1, x2, x3, . . . , xn. The CIA knows that the aircraft were numbered consecutively at the factory starting with and ending with , so that the total number of planes manufactured is 1 (e.g., if 17 and 29,
13. Consider a random sample X1, . . . , Xn from the pdf f(x; ) .5(1 x) 1 x 1 where 1 1 (this distribution arises in particle physics). Show that ˆ 3X is an unbiased estimator of .[Hint: First determine E(X) E(X).]
12. Suppose a certain type of fertilizer has an expected yield per acre of 1 with variance 2, whereas the expected yield for a second type of fertilizer is 2 with the same variance 2.Let S2 1 and S2 2 denote the sample variances of yields based on sample sizes n1 and n2, respectively, of the two
c. How would you use the observed values x1 and x2 to estimate the standard error of your estimator?d. If n1 n2 200, x1 127, and x2 176, use the estimator of part (a) to obtain an estimate of p1 p2.e. Use the result of part (c) and the data of part (d) to estimate the standard error of
b. What is the standard error of the estimator in part (a)?
a. Show that (X1/n1) (X2/n2) is an unbiased estimator for p1 p2. [Hint: E(Xi) nipi for i 1, 2.]
11. Of n1 randomly selected male smokers, X1 smoked filter cigarettes, whereas of n2 randomly selected female smokers, X2 smoked filter cigarettes. Let p1 and p2 denote the probabilities that a randomly selected male and female, respectively, smoke filter cigarettes.
10. Using a long rod that has length , you are going to lay out a square plot in which the length of each side is . Thus the area of the plot will be 2. However, you do not know the value of , so you decide to make n independent measurements X1, X2, . . . Xn of the length. Assume that each Xi has
9. Each of 150 newly manufactured items is examined and the number of scratches per item is recorded (the items are supposed to be free of scratches), yielding the following data:
8. 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 components 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
7.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 denote the
6. 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 article 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
5. As an example of a situation in which several different statistics 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
4. The article from which the data of Exercise 1 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 X1, . . . , Xm and the cylinder
3. 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
2. A sample of 20 students who had recently taken elementary statistics yielded the following information on brand of calculator owned (T Texas Instruments, H Hewlett Packard, C Casio, S Sharp):TTHTCTTSCH SSTHCTTTHTa. Estimate the true proportion of all such students who own a Texas
1. The accompanying data on flexural strength (MPa) for concrete 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 X and Y be independent standard normal random variables, 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) for a specified value of ?
A more accurate approximation to E[h(X1, . . . , Xn)] in Exercise 93 is h(1, . . . , n) 1 22 1 . . . 1 22 n Compute this for Y h(X1, X2, X3, X4) given in Exercise 93, and compare it to the leading term h(1, . . . , n).
Let X1, . . . , Xn be independent rv’s with mean values 1, . . . , n and variances 2 1, . . . , 2 n. Consider a function h(x1, . . . , xn), and use it to define a new rv Y h(X1, . . . , Xn). Under rather general conditions on the h function, if the is are all small relative to the corresponding
Let A denote the percentage of one constituent in a randomly selected rock specimen, and let B denote the percentage 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 X1 and X2 the two measured weights. Then X1 W E1 and X2 W E2, where E1 and E2 are the two measurement errors. Suppose that the Eis are independent of one another and of
a. Show that Cov(X, Y Z) Cov(X, Y) Cov(X, Z).b. Let X1 and X2 be quantitative and verbal scores on one aptitude exam, and let Y1 and Y2 be corresponding scores on another exam. If Cov(X1, Y1) 5, Cov(X1, Y2) 1, Cov(X2, Y1) 2, and Cov(X2, Y2) 8, what is the covariance between the two total
a. Let X1 have a chi-squared distribution with parameter 1(see Section 4.4), and let X2 be independent of X1 and have a chi-squared distribution with parameter 2. Use the technique of Example 5.21 to show that X1 X2 has a chi-squared distribution with parameter 1 2.b. In Exercise 71 of Chapter 4,
Suppose a randomly chosen individual’s verbal score X and quantitative score Y on a nationally administered aptitude examination have joint pdf f(x, y) {2 5(2x 3y) 0 x 1, 0 y 1 0 otherwise You are asked to provide a prediction t of the individual’s total score X Y. The error of
a. Use the general formula for the variance of a linear combination to write an expression for V(aX Y). Then let a Y/X, and show that 1. [Hint: Variance is always 0, and Cov(X, Y) X Y .]b. By considering V(aX Y), conclude that 1.c. Use the fact that V(W) 0 only if W is a
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
Refer to Exercise 58, and suppose that the Xis are independent with each one having a normal distribution. What is the probability that the total volume shipped is at most 100,000 ft3?
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 13 oz and 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 weeks
Let 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 and standard deviation .1. How many determinations are required if we wish the probability that the sample average is
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
We have seen that if E(X1) E(X2) . . . E(Xn) , then E(X1 . . . Xn) n. In some applications, the number of Xis under consideration is not a fixed number n 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
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 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
Let X1, X2, . . . , Xn be random variables denoting n independent bids for an item that is for sale. Suppose each Xi 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, . . . , Xn).
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 x 0, y 0, 20 x y 30 0 otherwisea. Draw the region of positive density and
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 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
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
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 X the sample average tensile
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 Xis are independent, normally distributed,
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
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
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.| Road 1 Road 2 Road 3 Expected value | 800 1000
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
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
If two loads are applied to a cantilever beam as shown in the accompanying drawing, the bending moment at 0 due to the loads is a1X1 a2X2.a. Suppose that X1 and X2 are independent rv’s with means 2 and 4 kips, respectively, and standard deviations .5 and 1.0 kip, respectively. If a1 5 ft and a2
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
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 time?
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 superexpress checkout.b. Calculate V(X1 X2). How does this compare to V(X1) V(X2)?
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,
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
Five automobiles of the same type are to be driven on a 300-mile trip. The first two will use an economy brand of gasoline, and the other three will use a name brand. Let X1, X2, X3, X4, and X5 be the observed fuel efficiencies (mpg) for the five cars. Suppose these variables are independent and
Let X1, X2, and X3 represent the times necessary to perform three successive repair tasks at a certain service facility.Suppose they are independent, normal rv’s with expected values 1, 2, and 3 and variances 2 1, 2 2, and 2 3, respectively.a. If 2 3 60 and 2 1 2 2 2 3 15, calculate
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 i E(Xi) and 2 i V(Xi), suppose that the mean values and standard deviations are as
Suppose the distribution of the time X (in hours) spent by students at a certain university on a particular project is gamma with parameters 50 and 2. Because is large, it can be shown that X has approximately a normal distribution. Use this fact to compute the probability that a randomly
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

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