411_hw06 - Stat 411 Homework 06 Due: Wednesday 02/29...

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Stat 411 – Homework 06 Due: Wednesday 02/29 Undergraduates may solve the “Graduate only” problem(s) for possible extra credit. 1. Let X 1 ,X 2 be iid with PDF f θ ( x ) = (1 ) e - x/θ , x > 0. (a) Let Y 1 = X 1 and Y 2 = X 1 + X 2 . Find the joint PDF f Y 1 ,Y 2 ( y 1 ,y 2 ), the marginal PDF f Y 2 ( y 2 ) and, finally, conditional PDF f Y 1 | Y 2 ( y 1 | y 2 ). (b) Calculate g ( y 2 ) = E θ ( Y 1 | Y 2 = y 2 ). Does your formula depend on θ ? (c) Find E θ [ g ( Y 2 )] and V θ [ g ( Y 2 )] and compare to E θ ( Y 1 ) and V θ ( Y 1 ). 2. Problem 7.4.5 on page 388. 3. Let X 1 ,...,X n iid Ber ( θ ); then T = n i =1 X i is a complete sufficient statistic. The goal is to estimate η = θ (1 - θ ). (a) Show that the “naive” estimator ˜ η = X 1 (1 - X 2 ) is unbiased. (b) Find the better estimator ˆ η = E η | T = t ) from the Rao–Blackwell Theorem. You may assume that 0 < t < n . (Hint: The random variable X 1 (1 - X 2 ) is either 0 or 1; it’s 1 if and only if
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