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Miller And Freunds Probability And Statistics For Engineers 9th Edition Richard A. Johnson - Solutions

5.17 Find μ for the distribution of the satisfactory service of Exercise 5.10.5.18 Show that μ 2 and, hence, σ2 do not exist for the probability density of Exercise 5.6.
5.16 Find μ and σ for the distribution of the phase error of Exercise 5.9.
5.15 Find μ and σ for the probability density obtained in Exercise 5.8.
5.14 Find μ and σ2 for the probability density of Exercise 5.4.
5.13 Find μ and σ2 for the probability density of Exercise 5.2.
5.12 Prove that the identity σ2 = μ 2 − μ2 holds for any probability density for which these moments exist.
5.11 At a certain construction site, the daily requirement of gneiss (in metric tons) is a random variable having the probability density f (x) = ⎧ ⎨ ⎩ 4 81 (x + 2)−(x+2)/9 for x > 0 0 for x ≤ 0 If the supplier’s daily supply capacity is 25 metric tons, what is the probability that this
5.10 The length of satisfactory service (years) provided by a certain model of laptop computer is a random variable having the probability density f (x) = ⎧ ⎨ ⎩ 1 4.5 e−x/4.5 for x > 0 0 for x ≤ 0 Find the probabilities that one of these laptops will provide satisfactory service for (a)
5.9 Let the phase error in a tracking device have probability density f (x) = cos x 0 < x < π/2 0 elsewhere Find the probability that the phase error is (a) between 0 and π/4; (b) greater than π/3.
5.8 Find the probability density that corresponds to the distribution function of Exercise 5.7. Are there any points at which it is undefined? Also sketch the graphs of the distribution function and the probability density.
5.7 If the distribution function of a random variable is given by 4 for x > 2 F(x)= 0 for x 2 find the probabilities that this random variable will take on a value (a) less than 3; (b) between 4 and 5.
5.6 Given the probability density f(x) = -00 < x < , find k. k 1+x for
5.5 With reference to the preceding exercise, find the cor- responding distribution function, and use it to deter- mine the probabilities that a random variable having the distribution function will take on a value (a) greater than 1.8; (b) between 0.4 and 1.6.
5.4 If the probability density of a random variable is given by f (x) = ⎧ ⎨ ⎩ x for 0 < x < 1 2 − x for 1 ≤ x < 2 0 elsewhere find the probabilities that a random variable having this probability density will take on a value (a) between 0.2 and 0.8; (b) between 0.6 and 1.2.
5.3 With reference to the preceding exercise, find the corresponding distribution function and use it to determine the probabilities that a random variable having this distribution function will take on a value (a) between 0.45 and 0.75; (b) less than 0.6.
5.2 If the probability density of a random variable is given by f (x) = (k + 2)x3 0 < x < 1 0 elsewhere find the value k and the probability that the random variable takes on a value (a) greater than 3 4 ; (b) between 1 3 and 2 3 .
5.1 Verify that the function of Example 1 is, in fact, a probability density.
4.98 A candidate invited for a visit has probability 0.6 of being hired. Let X be the number of candidates that visit before 2 are hired. Find (a) P ( X ≤ 4 ); (b) P ( X ≥ 5 ).
4.97 Suppose that the probabilities are 0.2466, 0.3452, 0.2417, 0.1128, 0.0395, 0.0111, 0.0026, and 0.0005 that there will be 0, 1, 2, 3, 4, 5, 6, or 7 polluting spills in the Great Lakes on any one day. Simulate the numbers of polluting spills in the Great Lakes in 30 days.
4.96 A manufacturer determines that a big screen HDTV set had probabilities of 0.8, 0.15, 0.05, respectively, of being placed in the categories acceptable, minor defect, or major defect. If 3 HDTVs are inspected, (a) find the probability that 2 are acceptable and 1 is a minor defect; (b) find the
4.95 The number of weekly breakdowns of a computer is a random variable having a Poisson distribution with λ = 0.2. What is the probability that the computer will operate without a breakdown for 3 consecutive weeks?
4.94 Records show that the probability is 0.00008 that a truck will have an accident on a certain highway. Use the formula for the Poisson distribution to approximate the probability that at least 5 of 20,000 trucks on that highway will have an accident.
4.93 The daily number of orders filled by the parts department of a repair shop is a random variable with μ = 142 and σ = 12. According to Chebyshev’s theorem, with what probability can we assert that on any one day it will fill between 82 and 202 orders?
4.92 With reference to Exercise 4.87, find the mean and the variance of the distribution of the number of microelectrodes made from glass tubing using (a) the probabilities obtained in that exercise; (b) the special formulas for the mean and the variance of a hypergeometric distribution.
4.91 Use the Poisson distribution to approximate the binomial probability b(1; 100, 0.02).
4.90 Find the mean and the standard deviation of the distribution of each of the following random variables (having binomial distributions): (a) The number of heads in 440 flips of a balanced coin. (b) The number of 6’s in 300 rolls of a balanced die.(c) The number of defectives in a sample of
4.89 With reference to Exercise 4.88, find the variance of the probability distribution using (a) the formula that defines σ2; (b) the special formula for the variance of a binomial distribution.
4.88 As can be easily verified by means of the formula for the binomial distribution, the probabilities of getting 0, 1, 2, or 3 heads in 3 flips of a coin whose probability of heads is 0.4 are 0.216, 0.432, 0.288, and 0.064. Find the mean of this probability distribution using (a) the formula that
4.87 In 16 experiments studying the electrical behavior of single cells, 12 use micro-electrodes made of metal and the other 4 use micro-electrodes made from glass tubing. If 2 of the experiments are to be terminated for financial reasons, and they are selected at random, what are the probabilities
4.86 If the probability is 0.90 that a new machine will produce 40 or more chairs, find the probabilities that among 16 such machines (a) 12 will produce 40 or more chairs; (b) at least 10 will produce 40 or more chairs; (c) at most 3 will not produce 40 or more chairs.
4.85 If the probability is 0.20 that a downtime of an automated production process will exceed 2 minutes, find the probability that 3 of 8 downtimes of the process will exceed 2 minutes using (a) the formula for the binomial distribution; (b) Table 1 or software.
4.84 An engineering student correctly answers 85% of all questions she attempts. What is the probability that the first incorrect answer was the fourth one?
4.83 Check whether the following can define probability distributions, and explain your answers. (a) f (x) = x 10 , for x = 0, 1, 2, 3, 4. (b) f (x) = 1 3 , for x = −1, 0, 1.(c) f (x) = (x − 1)2 4 , for x = 0, 1, 2, 3.
4.82 Determine whether the following can be probability distributions of a random variable that can take on only the values of 0, 1, and 2: (a) f (0) = 0.34 f (1) = 0.34 and f (2) = 0.34. (b) f (0) = 0.2 f (1) = 0.6 and f (2) = 0.2. (c) f (0) = 0.7 f (1) = 0.4 and f (2) = −0.1.
4.81 Refer to Exercise 4.80 and obtain the (a) mean; (b) variance; (c) standard deviation for the number of requests for conference rooms.
4.80 Upon reviewing recent use of conference rooms at an engineering consulting firm, an industrial engineer determined the following probability distribution for the number of requests for a conference room per half-day: x f(x) 0 .07 1 .15 2 .45 3 .25 4 .08 (a) Currently, the building has two
4.79 A manufacturer of smart phones has the following probability distribution for the number of defects per phone: x f(x) 0 .89 1 .07 2 .03 3 .01 (a) Determine the probability of 2 or more defects. (b) Is a randomly selected phone more likely to have 0 defects or 1 or more defects?
4.78 Depending on the availability of parts, a company can manufacture 3, 4, 5, or 6 units of a certain item per week with corresponding probabilities of 0.10, 0.40, 0.30, and 0.20. The probabilities that there will be a weekly demand for 0, 1, 2, 3,…, or 8 units are, respectively, 0.05, 0.10,
4.77 The probabilities that a quality control team will visit 0, 1, 2, 3, or 4 production sites on a single day are 0.15, 0.22, 0.35, 0.21, and 0.07. (a) Simulate the inspection team’s visits on 30 days. (b) Repeat the simulation of visits on 30 days a total of 100 times. Estimate the probability
4.76 Simulate tossing a coin. (a) For a balanced coin, generate 100 flips. (b) For a coin with probability of heads 0.8, generate 100 flips.
4.75 Using the same sort of reasoning as in the derivation of the formula for the hypergeometric distribution, we can derive a formula which is analogous to the multi- nomial distribution but applies to sampling without re- placement. A set of N objects contains aj objects of the first kind, a
4.74 Suppose the probabilities are 0.89, 0.09, and 0.02 that the finish on a new car will be rated acceptable, easily repairable, or unacceptable. Find the probability that, among 20 cars painted one morning, 17 have accept- able finishes, 2 have repairable finishes, and 1 finish is unacceptable.
4.73 As can easily be shown, the probabilities of getting 0, 11 1, or 2 heads with a pair of balanced coins are and 1 What is the probability of getting 2 tails twice, 1 head and 1 tail 3 times, and 2 heads once in 6 tosses of a pair of balanced coins?
4.72 Suppose that the probabilities are, respectively, 0.40, 0.40, and 0.20 that in city driving a certain kind of im- ported car will average less than 22 miles per gallon, anywhere from 22 to 25 miles per gallon, or more than 25 miles per gallon. Find the probability that among 12 such cars
4.71 Cumulative Poisson probabilities can be calculated using MINITAB. Dialog box: Calc > Probability Distribution > Poisson Choose Cumulative Distribution. Choose Input constant and enter 2. Type 1.64 in Mean. Click OK. Output: Poisson with mean = 1.64 x P(X
4.70 Poisson probabilities can be calculated using MINITAB.Dialog box: Calc > Probability Distribution > Poisson Choose Probability. Choose Input constant and enter 2. Type 1.64 in Mean. Click OK. Output: Poisson with mean = 1.64 x P(X = x) 2 0.260864 Find the Poisson probabilities for x = 2 and x
4.69 Use the formulas defining μ and σ2 to show that the mean and the variance of the Poisson distribution are both equal to λ.
4.68 Differentiating with respect to p on both sides of the equation ∞ x=1 p(1 − p) x−1 = 1 show that the geometric distribution f (x) = p(1 − p) x−1 for x = 1, 2, 3,... has the mean 1/p.
4.67 The number of flaws in a fiber optic cable follows a Poisson process with an average of 0.6 per 100 feet. (a) Find the probability of exactly 2 flaws in a 200- foot cable. (b) Find the probability of exactly 1 flaw in the first 100 feet and exactly 1 flaw in the second 100 feet.
4.66 The arrival of trucks at a receiving dock is a Poisson process with a mean arrival rate of 2 per hour. (a) Find the probability that exactly 5 trucks arrive in a two-hour period. (b) Find the probability that 8 or more trucks arrive in a two-hour period. (c) Find the probability that exactly 2
4.65 During an assembly process, parts arrive just as they are needed. However, at one station, the probability is 0.01 that a defective part will arrive in a one-hour period. Find the probability that (a) exactly 1 defective part arrives in a 4-hour span; (b) 1 or more defective parts arrive in a
4.64 Referring to Exercise 4.63, find the probability that the 15th gear in a day is the fourth to fail the test.
4.63 A company manufactures hydraulic gears, and records show that the probability is 0.04 that one of its new gears will fail its inspection test. What is the probability that the fifth gear in a day will be the first one to fail the test?
4.62 An automated weight monitor can detect underfilled cans of beverages with probability 0.98. What is the probability it fails to detect an underfilled can for the first time when it encounters the 10th underfilled can?
4.61 In a “torture test,” a light switch is turned on and off until it fails. If the probability that the switch will fail any time it is turned on or off is 0.001, what is the probability that the switch will fail after it has been turned on or off 1,200 times? Assume that the conditions
4.60 Environmental engineers, concerned about the effects of releasing warm water from a power plants’ cooling system into a Great Lake, decided to sample many organisms both inside and outside of a warm water plume. For the zoo-plankton Cyclops, they collect 100 cc of water and count the
4.59 A conveyor belt conveys finished products to the warehouse at an average of 2 per minute. Find the probabilities that (a) at most 3 will be conveyed in a given minute; (b) at least 2 will be conveyed in an interval of 3 minutes; (c) at most 20 will be conveyed during an interval of 5 minutes
4.58 A consulting engineer receives, on average, 0.7 requests per week. If the number of requests follows a Poisson process, find the probability that (a) in a given week, there will be at least 1 request; (b) in a given 4-week period there will be at least 3 requests.
4.57 The number of gamma rays emitted per second by a certain radioactive substance is a random variable having the Poisson distribution with λ = 5.8. If a recording instrument becomes inoperative when there are more than 12 rays per second, what is the probability that this instrument becomes
4.56 During inspection of the continuous process of making large rolls of floor coverings, 0.5 imperfections are spotted per minute on average. Use the Poison distribution to find the probabilities (a) one imperfection in 4 minutes (b) at least two in 8 minutes (c) at most one in 10 minutes.
4.55 In a factory, 8% of all machines break down at least once a year. Use the Poisson approximation to the binomial distribution to determine the probabilities that among 25 machines (randomly chosen in the factory): (a) 5 will break down at least once a year; (b) at least 4 will break down once a
4.54 Use the Poisson distribution to approximate the binomial probability b(3; 100, 0.03).
4.53 Use Table 2W or software to find (a) F(9; 12); (b) f (9; 12); (c) 12 k=3 f (k; 7.5).
4.52 Use Table 2W or software to find (a) F(4; 7); (b) f (4; 7); (c) 19 k=6 f (k; 8).
4.51 Use the recursion formula of Exercise 4.50 to calculate the value of the Poisson distribution with λ = 3 for x = 0, 1, 2,..., and 9, and draw the probability histogram of this distribution. Verify your results by referring to Table 2W or software.
4.50 Prove that for the Poisson distribution f (x + 1; λ) f (x; λ) = λ x + 1 for x = 0, 1, 2,....
4.49 Prove that (a) σ2 = E(X2) − μ2; (b) μ3 = μ 3 − 3μ 2 · μ + 2μ3.
4.48 The time taken by students to fill out a loan request form has standard deviation 1.2 hours. What does Chebyshev’s theorem tell us about the probability that a students’ time will be within 4 hours of the mean μ for all potential loan applicants?
4.47 Show that for 48 million draws from a fair deck of cards, the probability is at least 0.9375 that the pro- portion of spades drawn will fall between 0.24975 and 0.25025.
4.46 In 1 out of 22 cases, the plastic used in microwave- friendly containers fails to meet heat standards. If 979 specimens are tested, what does Chebyshev's theorem tell us about the probability of getting at most 25 or more than 64 containers that do not meet the heat standards?
4.45 Over the range of cylindrical parts manufactured on a computer-controlled lathe, the standard deviation of the diameters is 0.002 millimeter. (a) What does Chebyshev’s theorem tell us about the probability that a new part will be within 0.006 unit of the mean μ for that run? (b) If the 400
4.44 Construct a table showing the upper limits provided by Chebyshev’s theorem for the probabilities of obtaining values differing from the mean by at least 1, 2, and 3 standard deviations and also the corresponding probabilities for the binomial distribution with n = 16 and p = 1 2 .
4.43 Prove the formula for the mean of the hypergeometric distribution with the parameters n,a, and N, namely, μ = n · a N . [Hint: Make use of the identity k r=0 m r s k − r = m + s k which can be obtained by equating the coefficients of xk in (1 + x) m(1 + x) s and in (1 + x)
4.42 Find the mean and the standard deviation of the hypergeometric distribution with the parameters n = 3, a = 4, and N = 8 (a) by first calculating the necessary probabilities and then using the formulas which define μ and σ; (b) by using the special formulas for the mean and the variance of a
4.41 Find the mean and the standard deviation of the distribution of each of the following random variables (having binomial distributions):(a) The number of heads obtained in 676 flips of a balanced coin. (b) The number of 4’s obtained in 720 rolls of a balanced die. (c) The number of
4.40 If 95% of certain high-performance radial tires last at least 30,000 miles, find the mean and the standard deviation of the distribution of the number of these tires, among 20 selected at random, that last at least 30,000 miles, using (a) Table 1, the formula which defines μ, and the
4.39 With reference to Exercise 4.38, find the variance of the probability distribution using (a) the formula that defines σ2; (b) the computing formula for σ2; (c) the special formula for the variance of a binomial distribution.
4.38 As can easily be verified by means of the formula for the binomial distribution (or by listing all 16 possibilities), the probabilities of getting 0, 1, 2, 3, or 4 red cards in four draws from a fair deck of cards are 1 16 4 16 6 16 4 16 1 16 Find the mean of this probability distribution
4.37 Find the mean and variance of the binomial distribution with n = 6 and p = 0.55 by using (a) Table 1 and the formulas defining μ and σ2; (b) the special formulas for the mean and the variance of a binomial distribution.
4.36 Find the mean and the variance of the uniform probability distribution given by f (x) = 1 n for x = 1, 2, 3,..., n [Hint: The sum of the first n positive integers is n(n + 1)/2, and the sum of their squares is n(n + 1) (2n + 1)/6.]
4.35 Use the computing formula for σ2 to rework part (b) of the preceding exercise.
4.34 The following table gives the probabilities that a certain computer will malfunction 0, 1, 2, 3, 4, 5, or 6 times on any one day: Number of malfunctions: x 0123456 Probability: f (x) 0.17 0.29 0.27 0.16 0.07 0.03 0.0 Use the formulas which define μ and σ to find (a) the mean of this
4.33 Use the computing formula for σ2 to rework part (b) of the preceding exercise.
4.32 Suppose that the probabilities are 0.4, 0.3, 0.2, and 0.1 that there will be 0, 1, 2, or 3 power failures in a certain city during the month of July. Use the formulas which define μ and σ2 to find (a) the mean of this probability distribution; (b) the variance of this probability
4.31 Cumulative binomial probabilities can be calculated using MINITAB. Dialog Box: Calc > Probability Distribution > Binomial Choose Cumulative Distribution. Enter 7 in Number of trials and .33 in Probability of success. Choose Input constant and enter 2. Click OK. Output: Cumulative Distribution
4.30 Binomial probabilities can be calculated using MINITAB. Dialog box: Calc > Probability Distribution > Binomial Choose Probability. Enter 7 in Number of trials and .33 in Probability of success. Choose Input constant and enter 2. Click OK. Output: Probability Density Function Binomial with n =
4.29 Refer to Exercise 4.24 but now suppose there will be 75 units among which 45 will need to undergo a speed test and 30 will be tested for current flow. Find the probability that, among the four inspections assigned to the engineers, 3 will be speed tests and 1 will not, by using (a) the
4.28 A shipment of 120 burglar alarms contains 5 that are defective. If 3 of these alarms are randomly selected and shipped to a customer, find the probability that the customer will get one bad unit by using (a) the formula for the hypergeometric distribution; (b) the formula for the binomial
4.27 Among the 13 countries that an international trade federation is considering for their next 4 annual conferences, 6 are in Asia. To avoid arguments, the selection is left to chance. If none of the countries can be selected more than once, what are the probabilities that (a) all the
4.26 If 6 of 18 new buildings in a city violate the building code, what is the probability that a building inspector,who randomly selects 4 of the new buildings for inspection, will catch (a) none of the buildings that violate the building code? (b) 1 of the new buildings that violate the building
4.25 A maker of specialized instruments receives shipments of 24 circuit boards. Suppose one shipment contains 4 that are defective. An engineer selects a random sample of size 4. What are the probabilities that the sample will contain (a) 0 defective circuit boards? (b) 1 defective circuit board
4.24 Suppose that, next month, the quality control division will inspect 30 units. Among these, 20 will undergo a speed test and 10 will be tested for current flow. If an engineer is randomly assigned 4 units, what are the probabilities that (a) none of them will need a speed test? (b) only 2 will
4.23 Four emergency radios are available for rescue workers but one does not work properly. Two randomly selected radios are taken on a rescue mission. Let X be the number that work properly between the two. (a) Determine the probability distribution f (x) of X. (b) Determine the cumulative
4.22 Refer to Exercise 4.2. (a) Determine the cumulative probability distribution F(x). (b) Graph the probability distribution of f (x) as a bar chart and below it graph F(x).
4.21 A food processor claims that at most 10% of her jars of instant coffee contain less coffee than claimed on the label. To test this claim, 16 jars of her instant coffee are randomly selected and the contents are weighed; her claim is accepted if fewer than 3 of the jars contain less coffee than
4.20 A quality-control engineer wants to check whether (in accordance with specifications) 95% of the electronic components shipped by his company are in good working condition. To this end, he randomly selects 15 from each large lot ready to be shipped and passes the lot if the selected
4.19 A milk processing unit claims that, of the processed milk converted to powdered milk, 95% does not spoil. Find the probabilities that among 15 samples of powdered milk (a) all 15 will not spoil; (b) at most 12 will not spoil; (c) at least 9 will not spoil.
4.18 The probability that the noise level of a wide-band amplifier will exceed 2 dB is 0.05. Use Table 1 or software to find the probabilities that among 12 such amplifiers the noise level of (a) one will exceed 2 dB; (b) at most two will exceed 2 dB; (c) two or more will exceed 2 dB.
4.15 Voltage fluctuation is given as the reason for 80% of all defaults in nonstabilized equipment in a plant. Use the formula for the binomial distribution to find the probability that voltage fluctuation will be given as the reason for three of the next eight defaults. 4.16 If the probability
4.14 Rework the decision problem in Example 7, supposing that only 3 of the 20 hard drives required repairs within the first year.

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