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Solutions08 - Homework 8 1 X2 ^ Xi a E X = 2 implies that E...

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Homework 8 1. a. implies that θ 2 ) ( 2 = X E θ = 2 2 X E . Consider n X i 2 ˆ 2 = θ . Then ( ) ( ) θ θ θ θ = = = = = n n n n X E n X E E i i 2 2 2 2 2 2 ˆ 2 2 , implying that is an unbiased estimator for θ ˆ θ . b. , so 1058 . 1490 2 = i x 505 . 74 20 1058 . 1490 ˆ = = θ 3. a. We wish to take the derivative of , set it equal to zero and solve for p. ( ) x n x p p x n 1 ln ( ) ( ) ( ) p x n p x p x n p x x n dp d = + + 1 1 ln ln ln ; setting this equal to zero and solving for p yields n x p = ˆ . For n = 20 and x = 3, 15 . 20 3 ˆ = = p b. ( ) ( ) ( ) p np n X E n n X E p E = = = = 1 1 ˆ ; thus is an unbiased estimator of p. p ˆ c. ( ) 4437 . 15 . 1 5 = 4. a. ( ) 2 1 1 2 1 1 ) ( 1 0 + = + + = + = θ θ θ θ θ dx x x X E , so the moment estimator is the solution to θ ˆ 2 ˆ 1 1 + = θ X , yielding 2 1 1 ˆ = X θ . Since . 3 2 5 ˆ , 80 . = = = θ x b.
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