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Probability Statistics And Reliability For Engineers And Scientists 3rd Edition Bilal M. Ayyub, Richard H. McCuen - Solutions
3.21 Problem 3.15.find sc and find the probability of cco 3sc.
3.22 Problem 3.16.find sc and find the probability of cco 3sc.
3.23 Problem 3.17.find sc and find the probability of cco 3sc.
3.24 If po¼0.20, n¼4, find the probability of none, one, two, three, and four nonconforming units in the sample.
3.25 If po¼0.10, n¼5 find the probability for each of 0, 1, . . . ,5 nonconformining units in the sample.
3.26 If from a lot of seven clocks of which two are nonconforming, a random sample of two is drawn, find P(0), P(1) and P(2).
3.27 If from a lot of nine gauges, three of which are inaccurate, a random sample of three is drawn, find P(0), P(1), P(2) and P(3).
4.1. For some production process with which you are familiar, make a list of chance causes and of potential assignable causes.
4.2. Inspection of fiber containers for contamination, resulting from gluing, from samples of n¼25.Sample number 1 2 3 4 5 6 7 8 Fraction nonconfiguring, p 0.04 0.20 0.04 0.04 0.00 0.08 0.08 0.08 Sample number 9 10 11 12 13 14 15 16 Fraction nonconfiguring, p 0.08 0.12 0.00 0.04 0.04 0.00 0.08
4.3. For 21 production days the total nonconformities, c, found on the day’s production of 3000 switches ran as follows for December:30 56 47 86 44 23 16 64 80 54 73 65 76 69 53 58 30 91 90 36 57
4.4. A transmission main shaft bearing retainer carried specifications of 2.8341–2.8351 in. In a production run, measurements were made in 0.0001 in. from the ‘‘nominal’’ 2.8346 in.Thus, the specifications were 5 to þ5 in the coded units. The following data on averages and ranges were
5.1. For the experimental data in Example 1, see Sec.5.1.1, for samples 31–60, analyze by an np control chart under the case analysis of past data. How is control? What would the control lines be for the case control against standard po¼0.10. Are the results compatible with po¼0.10?
5.2. For the experimental data in Example 1, see Sec.5.1.1, for samples 61–90, analyze by a p control chart under the case analysis of past data. How is control? What would the control lines be for the case control against standard po¼0.03. Are the results compatible with po¼0.03?
5.3. For the packing nut data of Table 4.1, find the central line and control limits for a p chart, analyzing as past data. Comment on control. In March samples of 50 packing nuts gave the following counts of np¼d: 0, 0, 4, 0, 1, 4, 5, 0, 0, 7, 0, 0, 0, 0, 0, 0, 2, 3. Make an appropriate chart and
5.4. For the data plotted in Problem 4.2, complete the p chart by finding and drawing in the control lines. Comment.(The next 25 samples showed d¼1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1.)
5.5. The present problem is on auto carburetors inspected at the end of the assembly line for all types of nonconformities.These data are for January 27, and follow the data of Example 2, Sec. 5.1.1. For a long time, a goal of po¼0.02 had been set. Plot the given data, either p or np, and set
5.6. The data here shown are for 100% inspection of lots of malleable castings for cracks by magnaflux. The lot size varies considerably, so a p chart is used instead of an np chart. By some care, one can use only four separate n’s. Make the control chart and comment.Lot number Lot size n
5.7. For the data of Problem 5.6, do you think that the occurrences of cracks in castings would be independent, that is, if one casting has a crack, this defect is unrelated to whether the next one has a crack? Might this account for the lack of control noted? Would you still investigate points
5.8. In Problem 5.6, p-bar¼0.1064 was found by use of(5.8), that is, 1873=17,608. Do you think you would obtain the same p by adding the 24 p values and dividing by 24?This would be an unweighted average of p’s. If we substitute np for d in (6.8) we have (P np)=P n. Thus p is a weighted average
5.9. Samples of 39 articles each, from an optical company, were examined for breakage. Forty-three such samples over two months time yielded d¼2, 1, 1, 0, 2, 3, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 0, 1, 0, 1, 0, 2, 0, 1, 1, 0, 1, 0, 7, 10, 6, 1, 2, 2, 2, 1, 1, 0, 2, 2, 2, 5, 2. Analyze by an appropriate
5.10. For the data on total nonconformities found on daily production of 3000 switches (Problem 4.3), find the central line and control limits, analyzing as past data.Comment on control. Make chart if not previously done.
5.11. The following data were observed just 1 year before the data of Problem 4.3 at the start of the control chart program. Analyze as past data, the 25 production counts of total nonconformities in 3000 switches per day: 450, 454, 564, 369, 294, 358, 343, 227, 263, 248, 692, 314, 247, 521, 435,
5.12. The following data are for 29 samples each of 100-yd lengths of woolen goods. The woolen goods pass slowly across a table and nonconformities noted are marked by passing a bit of yarn of contrasting color through the goods.Nonconformities: 2, 0, 0, 1, 1, 2, 2, 1, 0, 1, 1, 0, 2, 1, 1, 5, 0, 3,
5.13. For the experimental data on ‘‘seeds’’ in Table 5.2, analyze samples 51–75 as past data by a c chart. Does the process show control, that is, homogeneity? Also analyze for control against the standard co¼8.
5.14. Follow the same procedure as for Problem 5.13 but use samples 76–100 and co¼2.
5.15. For large aircraft assemblies, there were 18 categories of nonconformities. (The data were taken subsequently to those in Table 4.1.) Analyze by a c chart for whatever nonconformities are assigned and comment.Plane Alignment Tighten Cello seals Foreign matter Replace Plug holes 1 5 91 3 26 2
5.16. The following were the number of breakdowns in insulation in 5000-ft lengths of rubber covered wire: 0, 1, 1, 0, 2, 1, 3, 4, 5, 3, 0, 1, l, 1, 2, 4, 0, 1, 1, 0. What were the central line and control limits for the number of breakdowns? What would be the central line and limits for the
5.17. The following data show daily averages of errors per truck over the daily production of about 125 trucks at final inspection. These data precede those of Table 5.3.Day Total errors, c Average errors, u Day Total errors, c Average errors, u Nov 21 Th 156 1.25 Dec 5 Th 175 1.40 Nov 22 F 198
7.1. For Problem 6.1 from the previous chapter, you are asked to estimate the process capability. Is it appropriate to use the process capability indices? Why?Calculate the Cp, Cpk and Ppk for the process. Compare these and discuss any differences among them.
7.2. For Problem 6.5 using the samples 41–60 from the previous chapter, you are asked to estimate the process capability. Discuss how you would estimate the short-term standard deviation of the process, e.g., would you exclude the out-of-control ranges?Why or why not? Calculate the Cp, Cpk and
7.3. For Problem 6.8 from the previous chapter, you are asked to estimate the process capability. Calculate the Cp, Cpk and Ppk for the process. Compare these and discuss any differences among them.
7.4. For Problem 6.9 from the previous chapter, you are asked to estimate the process capability? Calculate the Cp, Cpk and Ppk. What do find?
A shipping company records the conditions of cargoes at the delivery points as not damaged (ND), damaged (D), and partially damaged (PD). The following probability mass function was established for three shipping methods:Determine the marginal probability mass functions PX(x) and PY(y). If a cargo
An electronics company tests for the performance quality of its products by performing nondestructive thermal and magnetic-field tests. The products are classified as adequate (A), not adequate (NA), and partially adequate (PA). A sample of 100 produced the following results:Determine the joint
A poultry inspector visually examines (VE) slaughtered chickens on a processing line in a plant.Birds that exhibit some visual marks are examined further based on a detailed examination (DE)protocol. A sample of 1000 birds has the following pass (P) and fail (F) results:Determine the joint
A weld inspector VE weld seams of a ship produced according to the American of Shipping Classification Rules. Welds seems that exhibit some visual signs of poor quality (P) or medium quality(M) are examined further using a nondestructive testing method (NDE). A sample of 100 weld seams has the
The electric (E) and gas (G) energy consumption of a household in a residential subdivision follows a hypothetical joint probability density function as follows:where e and g are normalized to the respective maximum value for all the houses. Determine the constant c such that fEG(e,g) is a
For a simply supported beam with a concentrated load P at midspan and uniformly distributed load of intensity w over the entire span L, the maximum shear force (V) and the maximum bending moment (M) are given byFind the covariance of V and M, Cov(V, M), using first-order approximations. Assume P
The change in the length of a rod shown in the figure above and due to an axial force P is given bywhere L is the length of rod, P is the applied axial force, A is the cross-sectional area of rod, and E is the modulus of elasticity. If P and E are normally distributed with μP and σP and μE and
The ultimate moment capacity, M, of an under-reinforced concrete rectangular section is given byin which the following are random variables: As is the cross-sectional area of the reinforcing steel, fy is the yield stress (strength) of the steel, d is the distance from the reinforcing steel to the
For the following engineering formula:compute the first-order approximate mean and variance for the following two cases: (1) uncorrelated Xs, and (2) correlated Xs as functions of the mean and variances and correlation coefficients of the Xs. Y = X, X, X
For the following engineering formula:compute the first-order approximate mean and variance for the following two cases: (1) uncorrelated Xs, and (2) correlated Xs as functions of the mean and variances and correlation coefficients of the Xs. Y = XX2X1/2X1/2
For the following engineering formula:compute the first-order approximate mean and variance for the following two cases: (1) uncorrelated Xs, and (2) correlated Xs as functions of the mean and variances and correlation coefficients of the Xs. Y = XX1/2X-1/2
For the following engineering formula:compute the first-order approximate mean and variance for the following two cases: (1) uncorrelated Xs, and (2) correlated Xs as functions of the mean and variances and correlation coefficients of the Xs. Y-XXXX%
For the following engineering formula: Y = X exp(X2)
For the following engineering formula:compute the first-order approximate mean and variance for the following two cases: (1) uncorrelated Xs, and (2) correlated Xs as functions of the mean and variances and correlation coefficients of the Xs. Y = X, exp(X2)
Using the midsquare method with a seed of 9507, generate a sample of ten values (X) that follow X ~N(μ = 10, σ = 2). Assume σY = 3, μY = 20, and ρ = 0.5. Generate ten values of Y using Equation 6.154 with values of Z generated using the midsquare method with a seed of 2782. Compute the
Using the midsquare method with a seed of 2941, generate 30 uniform variates. Use these to compute the material costs X assuming X ~ N(μ = 150, σ = 20). Using the midsquare method with a seed of 7676, generate 30 uniform variates and use them to compute the labor costs (Y) assuming Y ~ N(μ =225,
For the reinforced concrete section described in Problem 6.22, determine the second-order mean and variance of the moment capacity by including the second-order term of Taylor series expansion(Equation 6.113) in the approximate expression for the mean and variance.
The random variable Y is normally distributed with mean and standard deviation μY and σY, respectively.The dependent random variable Y is defined as Y = ln(X)Show that the probability density function of X is a lognormal distribution (as given by Equation 5.23).
The random variable X is normally distributed with mean and standard deviation μX = 0 and σX = 1, respectively. A dependent random variable Y is defined as Y = X2 Show that the probability density function of Y is a chi-square distribution (as given by Equation 5.66).
The consumption of two types of manufacturing goods (X and Y) follows a hypothetical joint probability density function as follows:f x y c x y x y XY ( , ) = ( + ) < < < < 2 for 0 1 and 0 1 where x and y are normalized to the respective maximum value for all the houses. Determine the constant c
The consumption of two types of household goods (X and Y) follows a hypothetical joint probability density function as follows:fXY (x, y) = c(x + y ) < x < < y < 2 2 for 0 1 and 0 1 where x and y are normalized to the respective maximum value for all the houses. Determine the constant c such that
Determine the correlation coefficient ρXY for the random variables in Problem 6.10.
Determine the correlation coefficient ρXY for the random variables in Problem 6.9.
Determine the conditional expected values E(X|y) and E(Y|x) for the random variables in Problem 6.10.
Determine the conditional expected values E(X|y) and E(Y|x) for the random variables in Problem 6.9.
The following is a joint probability density function for two continuous random variables, X and Y:f x y cx y x y XY ( , ) = for 0 < < 1 and 0 < < 1 Determine the constant c such that fXY(x,y) is a legitimate joint density function. Evaluate the marginal density functions fX(x) and fY(y). Evaluate
The following is a joint probability density function for two continuous random variables, X and Y:fXY (x, y) = cxy for 0 < x < 1 and 0 < y < 1 Determine the constant c such that fXY(x,y) is a legitimate joint density function. Evaluate the marginal density functions fX(x) and fY(y). Evaluate the
Determine the conditional probability mass function PY|X(Y = NA) for all x values for the random variables in Problem 6.2.
Determine the conditional probability mass function PX|Y(X = PA) for all y values for the random variables in Problem 6.2.
Determine the conditional probability mass function PY|X(Y = sea) for all x values for the random variables in Problem 6.1.
Determine the conditional probability mass function PX|Y(X = ND) for all y values for the random variables in Problem 6.1.
Construct a transformation curve that transforms uniform variates on a scale from 0 to 1 to uniform variates on a scale from 10 to 20. Use the 30 variates of Problem 3.32 and the transformation curve to obtain 30 uniform variates on a scale from 10 to 20. Discuss the results of the transformation.
Construct a transformation curve that transforms uniform variates on a scale from 0 to 1 to uniform variates on a scale from 10 to 20. Use the 30 variates of Problem 3.31 and the transformation curve to obtain 30 uniform variates on a scale from 10 to 20. Discuss the results of the transformation.
Re-do Problem 3.67 using different random numbers and compare the results. Discuss any differences.
Use the rand function to generate 1000 uniform random numbers (0 to 1). Plot frequency histograms using cell widths of 0.1, 0.2, and 0.25. Comment on: (a) the degree to which the numbers follow a uniform distribution, and (b) the effect of cell size on the ability of the sample to appear to have a
Use the rand function to generate 30 uniform random numbers (0 to 1). Plot frequency histograms using cell widths of 0.1, 0.2, and 0.25. Comment on: (a) the degree to which the numbers follow a uniform distribution, and (b) the effect of cell size on the ability of the sample to appear to have a
An oil exploration company drills for oil in a new territory. The probability of finding oil per drill is 0.70. Assuming statistically independent drills, compute the following: (1) the probability of finding oil in one well based on five wells, (2) the probability of finding oil in at least two
A defense manufacturer submits quotes to supply guns. The probability of winning a job is 0.65.Assuming statistically independent jobs, compute the following: (1) the probability of winning one job out of five submitted quotes, (2) the probability of winning at least two jobs out of five submitted
The probability that a flood of a specified magnitude occurs in any 1 year is 0.05. What is the probability that in the next 10 years (a) exactly two such floods will occur, (b) no more than one such flood will occur, (c) no such floods will occur, (d) at least four floods will occur, and (e) at
A air defense system has an effectiveness value to kill a single incoming target by firing one missile of 0.75. Assume targets and kills are independent events. (a) What is the probability of killing an incoming target by shooting two missiles at it? (b) What are the probabilities killing an
A stock trader in an equity market picks 10 stocks each month for a pension plan with an average positive return, successful pick, probability of 0.75 per pick after 1 year of holding a position. What is the probability of getting (a) exactly five successful picks out of the 10, (b) exactly two
A fair coin is tossed 10 times. What is the probability of getting (a) exactly five heads, (b) exactly two tails, (c) no heads, (d) two or fewer heads, (e) five or more tails, and (f) at least two but not more than six heads?
Water samples are taken from deep ocean water to identify new micro-species based on DNA testing.The probability of a sample containing new micro-species is 0.2. Determine the following:a. What is the probability of having one sample containing new micro-species out of 10 samples?b. What is the
Piles are used to support the foundation of a structure. The failure probability of a pile during proof testing is 0.1. Determine the following:a. The probability of 3 failed piles out of 10 tested pilesb. The probability of no failures in 20 tested pilesc. The probability of 10 tested piles to
The vaccination of children entails a composite probability of side effects of 0.01 per vaccination per child. The side effect requires a health clinic to stockpile special medication for these cases of side effects. The health clinic vaccinates on the average 200 children per day. How many doses
The annual failure probability of an Internet switch is 0.01. In a network that includes 10 such switches that are assumed to be independent, what is the probability of no failures? Assuming that these switches do not exhibit aging, what is the probability of no failures in 5 years?
The probability of a flood in any 1 year is 0.1. In a 10-year period, what is the probability of (a) no floods, (b) two or fewer floods, (c) from one to three floods?
The power consumption of pumps where measured over a period of one day. Determine the mean, variance, standard deviation, COV, and skewness for the tested pumps based on the following measurements:{0.65,0.88,0.75,0.50,0.93,0.85,0.78,0.55,0.70,0.65}
Ten steel specimens were tested for their yield strength. Determine the mean, variance, standard deviation, COV, and skewness for the tested steel based on the following test results:{ 38, 36, 34, 37, 38, 39, 35, 38, 40, 36} ksi
Calculate the mean, variance, standard deviation, COV, and skewness for the following data sample: X={3, 5, 8, 6,8}
An online order fulfillment center uses ground and air shipping to deliver packages. Ground shipping takes a duration D in days to reach a customer; whereas air shipping takes a duration T in days to reach a customer. Historical records indicate that customers prefer ground shipping over air
A delivery truck operates from a warehouse to a department store, and its trip duration is a random variable D in minutes. Once it reaches the department store, it takes time T to unload. To simplify the problem, the random variables D and T are assumed to take on discrete values with associated
A severe rainstorm is defined as a storm with 8 in. or more of rain. The cumulative distribution function of X as the random variable defining the amount of rain from a severe rainstorm is(a) Computea. (b) Graph both the density and the cumulative functions. (c) Compute the mean, median, and
For the probability density function of Problem 3.43, determine the mean, variance, standard deviation, COV, and skewness.
For the probability density function of Problem 3.40, determine the mean, variance, and standard deviation.
For the probability mass function of Problem 3.29, determine the mean, variance, standard deviation, COV, and skewness.
For the probability mass function of Problem 3.28, determine the mean, variance, and standard deviation.
The duration of a construction activity was estimated to be in the range [2, 6] days with a most likely duration of 4 days. A construction engineer used the following symmetric density function to model the duration:Determine the necessary constants a and b to have a legitimate density function.
Find the values of k that are necessary to make the following a legitimate density function: f(x)-[kexp(-kr) for 0sx < 10 otherwise
Find the value k that is necessary to make the following a legitimate density function:Graph both the density and the cumulative functions. fx(x)= 10 [2exp(-kr) for 0x < otherwise
Find the value k that is necessary to make the following a legitimate density function:Graph both the density and the cumulative functions. fx(x)-[kr for Osxsl 1k for 1x2
Find the value k that is necessary to make the following a legitimate density function for constant a and b:Graph both the density and the cumulative functions. fx(x)=. [k for asxsb 10 otherwise
Prospects for bidding by a contractor are ranked according to multi-criteria for bidding preference in a descending order. The cumulative probability function of winning an opportunity with the rank x out of n opportunities is assumed cumulative as follows:Graph both the density and the cumulative
Prospects for bidding by a contractor are ranked in a descending order. The probability mass function of winning a prospect with the rank X out of n opportunities is assumed as follows i.e., equally likely:Graph both the density and the cumulative functions. Compute the mean and variance. Px(x)-
The following probability mass function is for a random variable of the number of customers serviced by a bank teller:Assuming the mid value of the range to represent the x value for the probability mass function, (a) compute the expected value of X, (b) compute the variance and standard deviation,
For the following probability mass function:(a) Compute the expected value of X, (b) compute the variance and standard deviation, (c) compute the probability that X ≥ 6. x 4 5 6 7 8 9 10 Px(x) 0.05 0.05 0.25 0.2 0.2 0.15 0.1
For the following probability mass function:(a) Compute the expected value of X, (b) compute the variance and standard deviation, (c) compute the probability that X ≥ 6. x 4 5 6 7 8 9 10 Px(x) 0.40 0.30 0.20 0.05 0.03 0.02 0.01
The number of years of drought in each decade is as follows:Show the sample mass function and cumulative function for the number of drought years in a decade.What is the sample space? 1900 to 1909 1910 to 1919 1920 to 1929 1930 to 1939 1940 to 1949 1950 to 1959 1960 to 1969 1970 to 1979 1980 to
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