Final_Exam_2006 - PROF HONG FALL 2006 ECONOMICS 619 FINAL...

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PROF HONG FALL 2006 ECONOMICS 619 FINAL EXAM Notes: (1) This is a closed book/notes exam. There are 7 questions, with a total of 100 points; (2) you have 150 minutes; (3) suggestion: have a look at all problems and °rst solve the problems you feel easiest; (4) good luck! 1. [15 pts] Suppose Y = ° + ±X + j X j "; where E ( X ) = 0 ; var ( X ) = ² 2 X ; E ( " ) = 0 ; var ( " ) = ² 2 " ; and " and X are independent. Both ° and ± are constants. (a) [5 pts] Find E ( Y j X ) : (b) [5 pts] Find var ( Y j X ) : (c) [5 pts] Show cov ( Y; X ) = 0 if and only if ± = 0 : 2. [10 pts] Suppose f X i g n i =1 is an i.i.d. N ( ³; ² 2 ) random sample, where both ³ and ² 2 are unknown parameters. Contruct an unbiased estimator for ´ = ³ 2 , and justify it is unbiased. 3. [15 pts] (a) Suppose X n = f X i g n i =1 is an i.i.d. random sample with a population probability density f ( x; µ ) : Is X n a su¢ cient statistic for µ ? Give your reasoning. (b) Suppose X n = f X i g n i =1 is an i.i.d. random sample from a N ( µ; µ ) population, where µ is unknown. Find a su¢ cient statistic for µ . Give your reasoning.
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