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hw12 - duced by a certain semiconductor firm are...

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ORIE 3500/5500 – Engineering Probability and Statistics II Fall 2010 Assignment 12 Problem 1 Find the maximum likelihood estimator of the unknown parameter θ when X 1 , X 2 , . . . , X n is a sample from the distribution whose density function is f X ( x ) = 1 2 e -| x - θ | , -∞ < x < . Hint: If you run into difficulties, consider separately the cases where n is odd and n is even. Problem 2 For the sample in Problem 1 find the moment estimator of the parameter θ . Problem 3 Suppose that X 1 , . . . , X n are normal with mean μ 1 ; Y 1 , . . . , Y n are normal with mean μ 2 ; and W 1 , . . . , W n are normal with mean μ 1 + μ 2 . Assuming that all 3 n random variables are independent, with a common variance, find the maximum likelihood estimators of μ 1 and μ 2 . Problem 4 The functional lifetimes in hours of computer chips pro-
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Unformatted text preview: duced by a certain semiconductor firm are exponentially distributed with mean 1 /λ . Suppose that the prior distribution on λ is the Gamma distribution with density function g ( x ) = e-x x 2 2 , < x < ∞ . Determine the two Bayes estimates of λ . Problem 5 Suppose that X 1 ,...,X n is a sample from a normal distribution with 0 mean, and unknown variance θ = σ 2 . We postulate that the prior density of θ is the standard exponential density. Find the Bayes estimator of the unknown variance that maximizes the posterior density. This assignment is NOT to be turned in. 1...
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