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CEE 304 – UNCERTAINTY ANALYSIS IN ENGINEERING 2005 Final Examination 2:00 – 4:30 PM, Friday, December 16, 2005 Exam is open book and open notes. It lasts 150 minutes and there are 150 points. SHOW WORK! 1. (10 pts – 2 pts each) Short answer and true/false questions. WRITE ANSWERS IN EXAM BOOK. (a) True or False : The joint density function f XY (x,y) can always be obtained as the product f X (x)f Y (y). (b) Given two independent events A and B, the general formula needed to compute P(A B) in terms of P(A) and P(B) is _______________________. (c) The sample average computed with n observations has a standard error of ___________. (d) True or False : By the Central Limit Theorem, the estimator x of the mean is always normally distributed, assuming the sample is large enough. (e) True or False : Maximum likelihood estimators are always unbiased. 2. (6 pts) A charcoal filter can remove chlorinated hydrocarbons from drinking water. An environmental engineer collected 12 independent samples of water that passed through such a filter. The average of the 12 observed removal rates was x = 99.27% with sample standard deviation s = 0.25%. Assuming that the measurements are normally distributed about the true removal rate μ , (a) What is 99% confidence interval for μ ? (b) Does the interval computed above indicate that precise information about the mean μ is available? 3. (12 pts) This time of year there are Christmas lights everywhere. The bulbs seem to burn out at random times. Suppose that the system of lights decorating the home could be described by a Poisson Process, wherein the arrival rate of failures is 0.2 per hour. (a) What is the probability that no lights fail in 3 hours of operation? (b) During a week (50 hours of operation), what is the mean and variance of the number of failures that will occur? (c) What is the mean and variance of the time (operational) until the system experiences 4 failures? (d) What is the probability 10 hours (operational hours) pass before 2 lights fail?
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CEE 304 – UNCERTAINTY ANALYSIS IN ENGINEERING 2005 Final Examination 2:00 – 4:30 PM, Friday, December 16, 2005 4. (16 pts) A structural engineer is concerned with the possible wind force on the side of a tall building. The total force F depends on the wind speed W, and the angle of the wind which determines the effective area A, such that F = W*A. The mean and standard deviation of W are 160 and 120. [Thus, the mean and standard deviation of ln(W) are 4.85 and 0.668.] The effective area A has median 50 and coefficient of variation 0.3, Assuming that W and A are independent random variables, and are both lognormally distributed, what are the median, mean, and standard deviation of the total force F on the structure? 5. (4 pts) A track student looked up the scores for the fastest times nationally to run 100 meters in each of the last 20 years. What is an appropriate distribution to describe these 20 values? Why?
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