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1 Exercises 1. A microcomputer system consists of a microprocessor CPU chip and a random access memory chip. The CPU is selected from a lot of 100 of which 10 are defective and the memory chip is selected fro a lot of 300 of which 15 are defective. Define A to be the event the selected CPU chip is defective and B the selected memory chip is defective. Since the two chips are selected from different lots, we may expect the events A and B to be independent. Could you check that? 2. There are 4 nonconforming capacitors in a lot of 25. A sample of 6 capacitors is taken. What is the probability of finding 2 nonconforming capacitors in the sample? Also find the mean and the standard deviation. 3. A random sample of 5 shafts is chosen from a Normal population with a mean diameter of 45mm and standard deviation 0.2mm. What is the probability that the mean diameter of a sample is less than 44.92mm? 4. The difference in the proportion of nonconforming components produced by 2 processes is to be established. A random sample of 96 parts taken from one process produced 6 non conforming parts and a sample of 120 parts taken from the second process had 8 nonconforming parts. Find the 95% confidence interval for the difference in proportion of non conforming parts. 5. A random sample of 12 silicon wafers has an average thickness of270µm. The standard deviation is known to be 8µm. Determine a twosided 99% CI interval for the thickness mean. 6. A random sample of 140 silicon wafers has 12 nonconforming units. Estimate the process fraction that are nonconforming and construct a two sided 95% CI for the true proportion that are non conforming. 7. A experiment is set up as H 0 : µ = µ 0, H 1 : µ µ 0 and is tested at a level of significance of 0.1. The sample size is 15. Determine the probability of rejecting H 0 if the true µ has shifted 0.25 standard deviation to the right of µ A system is made up of n independent components with reliability (Probability of working correctly) R i for the component C i. We will assume that failure events of components are mutually independent. Several organizations are possible : (a) A series system is one in which all components are so interrelated that the entire system will fail if any one of its components fails. Compute the reliabilty of a series system. 1
2 (b) On the other hand, a parallel system is one that will fail only if all its components fail (parallel redundancy). Compute the reliability of a parallel system. The following pictures describe respectively a series system and a parallel system. C 1 C 2 C i C n C 1 C 2 C i C n (c) Partial redundancy systems : the system has n components, each of them with a reliability R but the systems works if at least k components among n are working. Compute the reliability of such a system. 2
3 9. Let a telecommunication network linking different cities : V 2 V 3 V 1 V 6 V 4 V 5 Let R be the probability of success (no failure) of an interconnection (duplex link). We suppose that 2 cities V i and V j are connected if there is at least one path linking V i à V j with all its interconnections working. a) Give the probability that V 2 and V 6 can communicate correctly. Application for R = 0.99 (= ). b) Study the problem for V 1 and V 6. Solve this problem with conditional probabilities. Application for R = The following system is working if there is at least one working path from A to B. A C 1 C 2 C 4 B C 3 C 5 X i = C i is working ; R i = P(X i ) X = the system is working ; R = P(X) Compute R. 3
4 11. A transistor failure is modeled using a constant failure rate of per hour. Find the reliability of the transistor after 6000 hours of operation. Also determine the MTTF(Mean Time to Failure). 12. If a product reliability of 0.9 is to be achieved after 8000 hours of operation, determine the failure rate assuming an exponential distribution. 13. The life (hours) of a component is modeled using lognormal distribution. The 2 parameters for the distribution is given as mean µ = 5.5 and standard deviation σ = 1.6 Detremine the reliability of the component at 1000 hours and its MTTF. 14. A television camera focus system has 8 components in series. Each component failure has an exponential distribution with a failure rate of 40 per 10 6 hours. Determine the reliability at the end of 5000 hours of operation. Also calculate the MTTF for the system. If a reliability of 0.95 is desired after 5000 hours for the television camera, what should be the failure rate for each component? 15. A system has 3 components connected in parallel. The reliabilities of the components are 0.92,0.88 and 0.95 respectively. Determine the system reliability. If these reliabilities are at time 2000 hours of operation what is the MTTF of the system? (Assume exponential distribution for reliabilities) 16. A system has 5 components with system success defined as 4 out of 5. The reliability of each component is R = 0.9 at time T = 1. Detremine the system reliability and the MTTF. 17. A system has one basic unit and 2 standby units. The failure rate for each component is per hour. Find the system reliability at 400 hours of operation. Also determine the MTTF of the system. 18. A population of components is decribed by its life distribution (in hours) F(t) = 1 + ( t) 1 What is the probability that a new unit will fail by 1000 hours? by 4000 hours? between 1000 and 4000 hours? What proportion of these components will last more than 9000 hours? If we use 150 of them, how many do we expect to fail in the first 1000 hours?in the next 3000 hours? Derive the failure rate and calculate it at 10,100, 1000 and hours. Give the last failure rate in both PPM/K(per million per thousand hours) 19. The company Lifetime light bulb makes an incandescent filament that they believe does not wear out during an extended period of 4
5 normal use. They want w to guarantee it for 10 years of operation. To estimate the cost of such a guarantee, the Quality Departement takes a sample of 100 and realizes some accelerated tests (they have been given 3 months for that). Fortunately, the engineer who has to come with a test plan has a verified way of stressing light bulbs (using higher than normal voltages) which can simulate a month of typical use by a buyer in 1 hour of laboratory testing. He is able to take a random sample of 100 bulbs and test them all until failure in less than 3 months. The test engineer wants to use an exponential distribution for its bulbs failures based on his past experience. The sample data are the following : We have grouped the data in classes : lifetime in hours frequency plus de 440h 4 (a) Draw the histogram of the distribution. (b) Give the mode, the median, 1st and 3rd quartiles. (c) We suppose that the underlying distribution is exponential with parameter λ. Estimate the MTTF and λ. (d) Find a 95% CI interval for the true mean of the distribution. (e) Give an estimate of the reliability at 10 years. 5
6 (f) You want to check if the distribution is really exponential by performing a test. Give the hypothesis and perform a goodness of fit test. 20. The following data were collected in a test and are used to attempt a lognormal fit. The sample size is 15. Determine the lognormal parameters. The failure times in hours are 37.2, 39.2, 50.3, 52.6, 54.2, 66.0, 67.6, 70.9, 99.6, 106.5, 114.6, 128.7, 141.6, 197.3, A failure terminated test was performed on 15 gyro units without replacement. The test was terminated after 5 failures. The failure times in hours are 642,674,705,722, 732. Determine the 95% CI for the mean life using the exponential timetofailure model. 6
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