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Statistics Homework Solutions 100

# Statistics Homework Solutions 100 - 100 CHAPTER 4(c We...

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100 CHAPTER 4 (c) We denote the mean of μ 3 by E μ 3 and the variance of μ 3 by V μ 3 . E μ 3 μ X 1 μ X 2 4 μ μ 4 μ 2 . The bias of μ 3 is E μ 3 μ μ 2 μ μ 2 . The variance of μ 3 is V μ 3 σ 2 σ 2 16 σ 2 8 . The mean squared error of μ 3 is the sum of the variance and the square of the bias, so MSE μ 3 σ 2 8 μ 2 2 2 μ 2 1 8 . (d) μ 3 has smaller mean squared error than μ 1 whenever 2 μ 2 1 8 1 2 . Solving for μ yields 1 2247 μ 1 2247. (e) μ 3 has smaller mean squared error than μ 2 whenever 2 μ 2 1 8 5 9 . Solving for μ yields 1 3123 μ 1 3123. 3. The probability mass function of X is f x ; p p 1 p x 1 . The MLE is the value of p that maximizes f x ; p , or equivalently, ln f x ; p . d dp ln f x ; p d dp ln p x 1 ln 1 p 1 p x 1 1 p 0. Solving for
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