CEE_304_2nd_Exam2005 - CEE 304 Uncertainty Analysis in...

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CEE 304 – Uncertainty Analysis in Engineering Prelim #2 November 11, 2005 You may use the text, your notes, and calculators. There are 50 points in total. You have 50 minutes. 1. (12 pts) Consider the distribution with pdf: r Y y ) 1 r ( ) y ( f + = for 0 < y < 1, and 0 otherwise for which E[Y] = ) 2 r /( ) 1 r ( + + and Var[Y] = ] ) 2 r )( 3 r [( ) 1 r ( 2 + + + . Suppose one had a sample of independent observations y 1 , y 2 , …, y n . (a) What is the maximum likelihood estimator of r? (b) What is the method of moments estimator of r? (c) Is ! " = = n 1 i 2 i n 1 2 ) X X ( S an unbiased estimator of σ 2 ? (d) Why might a minimum MSE estimator be preferred over a MVUE estimator? 2. (8 pts) A transportation engineer wants to estimate the mean weight of passenger cars on Green Street in the morning. She set up a portable scale and weighed 15 randomly selected cars. She computed an average weight of 2570 pounds with a sample standard deviation of 460 pounds. (a) Construct a 95% confidence interval for the true mean weight of a passenger car. (b) The engineer plans to use this same sampling procedure on 10 other roads which her supervisor will specify. What is the probability that all 10 of the intervals that will be generated will contain the true means for the respective roads? (c) What is the probability that the particular interval constructed with the data collected for Green Street contains the true mean for Green Street?
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