solpm2

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

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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
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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