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Practice Midterm I

# Practice Midterm I - based on X 1 X n(b(10 pts How large...

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STA131C, Spring 2007 Prof. W. Polonik Practice Exercises I 1. Let X 1 , . . . , X n iid Gamma( α, λ ) with pdf f ( x ; α, λ ) = 1 Γ( λ ) λ α x α - 1 e - λx , x > 0 , where α, λ > 0 are the parameters. (a) Assume that α is known. Find the MLE for λ. (b) Is the MLE from (a) minimal sufficient? Justify your answer. 2. (20 pts) Let X 1 , . . . , X n i.i.d. N ( μ, 9) . (a) (10 pts) Construct the UMP-test at 5%-level for H 0 : μ 0 versus H 1 :
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Unformatted text preview: , based on X 1 , . . ., X n . (b) (10 pts) How large has n to be, such that the UMP-test from part (a) has power at least 0.99 at μ = 2 . 3. Let X 1 , . . ., X n ∼ iid Bernoulli( p ) . Let T n = ∑ n i =1 X i . (a) Find the estimator S n = E( X 1 | ∑ n i =1 X i ) . (b) Explain why the result of part (a) is not a surprise....
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