# Pm2sol - STA 103 Practice Midterm II Solutions 1(a ^ ^ ^ ^ ^ ^ TRUE Criterion is choose estimator"1 over 2 if mse"1 < mse 2 otherwise choose 2

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1 STA 103 Practice Midterm II Solutions 1 (a) TRUE. Criterion is: choose estimator ˆ " 1 over ˆ 2 if mse( ˆ 1 ) < mse( ˆ 2 ), otherwise choose ˆ 2 over ˆ 1 . For unbiased estimators, bias is equal to zero, so the criterion is equivalent to: choose estimator ˆ 1 over ˆ 2 if Var( ˆ 1 ) < Var( ˆ 2 ), otherwise choose ˆ 2 over ˆ 1 . (b) FASLE. Central limit theorem tells us that as the sample size n increases, the probability distribution of the sample mean X approaches a normal distribution. 2 (a) Since X and Y are independent, P(X,Y) = P(X)P(Y). Then, p X,Y (1,0) = p X (1) p Y (0) = (.3)(.3) = .09, and so on. The joint distribution of X and Y is: Y 0 1 X 1 .09 .21 2 .12 .28 3 .09 .21 (b) X 2 = EX 2 # ( EX ) 2 = X 2 p ( X ) x \$ # X p ( X ) x \$ ( ) 2 = 1(.3) + 4(.4) + 9(.3) # (.3 + 2(.4) + 3(.3)) 2 = 4.6 # 2 2 = 0.6 Y 2 = EY 2 # ( EY ) 2 = Y 2 p ( Y ) y \$ # Y p ( Y ) y \$ ( ) 2 = 0 2 (.3) + 1 2 (.7) # (0(.3) + 1(.7)) 2 = .7 # .7 2 = 0.21 (c) Since X and Y are independent, ρ X,Y = 0. 3 (a) The marginal distributions of X and Y are: X P(X) Y P(Y) -1 .2 1 .45

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## This note was uploaded on 05/18/2008 for the course STATS 103 taught by Professor Sen during the Winter '08 term at UC Davis.

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Pm2sol - STA 103 Practice Midterm II Solutions 1(a ^ ^ ^ ^ ^ ^ TRUE Criterion is choose estimator"1 over 2 if mse"1 < mse 2 otherwise choose 2

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