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HWsolutions2

# HWsolutions2 - IE121 HW#2 Solutions 7-2 E E ^ 1 1 E X1 7 1...

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IE121 HW#2 Solutions 7-2. E ( 29 [ ] μ μ Θ = = = + + + = ) 7 ( 7 1 )) ( 7 ( 7 1 ) ( ) ( ) ( 7 1 ˆ 7 2 1 1 X E X E X E X E E ( 29 [ ] μ μ μ μ Θ = + - = + + = ] 2 [ 2 1 ) ( ) ( ) 2 ( 2 1 ˆ 7 6 1 2 X E X E X E a) Both 1 ˆ Θ and 2 ˆ Θ are unbiased estimates of μ since the expected values of these statistics are equivalent to the true mean, μ . b) V ( 29 ( 29 2 2 7 2 1 2 7 2 1 1 7 1 ) 7 ( 49 1 ) ( ) ( ) ( 7 1 7 ... ˆ σ σ Θ = = + + + = + + + = X V X V X V X X X V 7 ) ˆ ( 2 1 σ Θ = V V ( 29 ( 29 )) ( ) ( ) ( 4 ( 4 1 ) ( ) ( ) 2 ( 2 1 2 2 ˆ 4 6 1 4 6 1 2 4 6 1 2 X V X V X V X V X V X V X X X V + + = + + = + - = Θ = ( 29 1 4 4 2 2 2 σ σ σ + + = 1 4 6 2 ( ) σ 2 3 ) ˆ ( 2 2 σ Θ = V Since both estimators are unbiased, the variances can be compared to decide which is the better estimator. The variance of 1 ˆ Θ is smaller than that of 2 ˆ Θ , 1 ˆ Θ is the better estimator. 7-4. Since both estimators are unbiased: Relative Efficiency = 21 2 7 / 3 7 / ) ˆ ( ) ˆ ( ) ˆ ( ) ˆ ( 2 2 2 1 2 1 = = = σ σ Θ Θ Θ Θ V V MSE MSE 7-8. n 1 = 20, n 2 = 10, n 3 = 8 Show that S 2 is unbiased: ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 2 3 2 2 2 1 2 3 2 2 2 1 2 3 2 2 2 1 2 38 38 1 8 10 20 38 1 8 10 20 38 1

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HWsolutions2 - IE121 HW#2 Solutions 7-2 E E ^ 1 1 E X1 7 1...

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