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m207sol - Midterm Examination # 2 Sta 113: Probability and...

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Unformatted text preview: Midterm Examination # 2 Sta 113: Probability and Statistics in Engineering Thursday, 2007 Oct. 25, 1:15 2:30 pm This is a closed-book exam so do not refer to your notes, the text, or any other books (please put them on the floor). You may use a single sheet of notes or formulas and a calculator, but materials may not be shared. A formula sheet and two blank worksheets are attached to the exam. You must show your work to get partial credit. Even correct answers will not receive full credit without justification. Please give all numerical answers to at least two correct digits or as exact fractions reduced to lowest terms . Write your solutions as clearly as possible and make sure its easy to find your answers (circle them if necessary), since you will not receive credit for work that I cannot understand or find. Good Luck! If you find a question confusing please ask me to clarify it. Cheating on exams is a breach of trust with classmates and faculty, and will not be tolerated. After completing the exam please acknowledge the Duke Honor Code with your signature below: I have neither given nor received unauthorized aid on this exam . Signature: Print Name: 1. 2. 3. 4. 5. Total: Name: Sta 113 Problem 1 : For each problem answer true or false and state WHY. i Given the Cauchy distribution Cauch( x ) = p ( x ) = 1 (1 + x 2 ) we draw x 1 ,...,x n i.i.d. Cauch. What does the Central Limit Theorem say about 1 n n X i =1 x i . False. The CLT does not hold in this case since this distribution does not have bounded variance or mean. ii Given a uniform prior the Bayes estimate and the maximum a poste- riori (MAP) estimate are equivalent. False. The MLE and the MAP are equivalent under this setting but the Bayes estimate and MAP may not since there is no gaurantee that the posterior will be symmetric, the condition under which the MAP and the Bayes estimate are equivalent. iii The minimum variance unbiased estimator is always the best estima- tor. False. There may exist a slightly biased estimator with far less variance than the MVUE. iv Given two random variables X and Y that are independent var( X- Y ) = var( X )- var( Y ) . False. var( X- Y ) = var(1 X +- 1 Y ) = 1 2 var( X ) + (- 1) 2 var( Y ) . v The maximum likelihood estimator for i.i.d. exponential random vari- ables is biased. True. The MLE estimator of is = 1 x . Fall 2007 Page 1 of 10 Oct. 25, 2007 Name: Sta 113 This is not unbiased since given data ( x 1 ,..,x n ) E x 1 ,...,x n [ ] = E x 1 ,...,x n [ 1 x ] 6 = . Fall 2007 Page 2 of 10 Oct. 25, 2007 Name: Sta 113 Problem 2 : The following joint distribution is on three exponentially decaying random variables p ( x,y,z ) = x y z exp[- ( x x + y y + z z )] x,y,z, where x,y,z are random variables describing the amount of radiation emit- ted from a radioactive source constructed by Lex Luther....
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This note was uploaded on 01/17/2010 for the course STA 113 taught by Professor Staff during the Fall '08 term at Duke.

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m207sol - Midterm Examination # 2 Sta 113: Probability and...

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