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solpm2 - Midterm Examination # 1 Sta 113: Probability and...

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Midterm Examination # 1 Sta 113: Probability and Statistics in Engineering Thursday, 2006 Oct. 26, 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 Foor). You may use a single sheet of notes or formulas and a calculator, but materials may not be shared. A formula sheet and four 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 justi±cation. 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 it’s easy to ±nd your answers (circle them if necessary), since you will not receive credit for work that I cannot understand or ±nd. Good Luck! If you ±nd 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. /15 2. /25 3. /15 4. /25 5. /10 6. /10 Total: /100
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Name: Sta 113 Problem 1 : For each problem answer true or false and state WHY. ( i The Central Limit Theorem states that given n random variables ( X 1 , ..., X n ) drawn i.i.d. from a distribution with mean μ and vari- ance σ 2 the average of the random variables denoted as ¯ X = n i =1 x i n is distributed approximately as a Gaussian with mean μ and variance σ 2 n if and only if the original distribution of is Gaussian. False. The CLT will hold for any distribution with ±nite means and variance, for example the uniform over [0 , 20]. ii The maximum likelihood estimator is always a unbiased estimator. False. The exponential distribution is an example. The MLE is in this case ˆ λ = 1 / ¯ X. This is not unbiased. iii The minimum variance unbiased estimator is always the best estima- tor or the principle of minimum variance unbiased estimator is always the best idea. False. There can exist an estimator that is slightly biased but has variance much smaller than the variance of unbiased estimators with minimum variance. v If X 1 , .... , X n are independent random variables and ( a 1 , ..., a n ) are real numbers then V ( a 1 X 1 + ... + a n X n ) = a 1 V ( X 1 ) + ... + a n V ( X n ) . False. V ( a 1 X 1 + ... + a n X n ) = a 2 1 V ( X 1 ) + ... + a 2 n V ( X n ) . Fall 2006 Page 1 of 10 Oct. 26, 2006
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Name: Sta 113 Problem 2 : a You are given the following joint density f ( x, y ) = 1 2 πσ x σ y exp - " σ 2 y ( x - μ x ) 2 2 σ 2 x σ 2 y + σ 2 x ( y - μ y ) 2 2 σ 2 x σ 2 y #! . Are the random variables
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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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solpm2 - Midterm Examination # 1 Sta 113: Probability and...

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