hw2_sol - Homework 2 Solution 10-2 We know that E ( X i ) =...

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Homework 2 Solution 10-2 We know that ( ) i E X μ = , which leads to [ ] [ ] 1 1 2 7 2 1 6 4 1 ˆ ( ) ( ) ( ) ( ) 7 1 ˆ ( ) 2 ( ) ( ) ( ) 2 E E X E X E X E E X E X E X θ = + + + = = - + = m So, both are unbiased estimators. We also know that 2 ( ) i Var X σ = , which means 2 2 2 1 1 7 2 2 2 2 2 1 6 4 1 1 ˆ ( ) ( ) ( ) 7 7 7 2 1 1 3 ˆ ( ) ( ) ( ) ( ) 2 2 2 2 Var Var X Var X Var Var X Var X Var X = + + = - = + + = m 2 2 1 1 ˆ ˆ ( ) ( ) ( ) 7 MSE Var bias = + = 2 2 2 2 3 ˆ ˆ ( ) ( ) ( ) 2 MSE Var bias = + = So, 1 ˆ is the better estimator of , because its MSE is smaller 10-7 For the Poisson distribution,
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