notes-11-29

notes-11-29 - M4056 Take-Home Test Answers November 29,...

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Unformatted text preview: M4056 Take-Home Test Answers November 29, 2010 Problem 4. a) ( x i ) = P ( X = x i | H ) P ( X = x i | H 1 ) . Therefore, ( x 1 ) = . 2 / . 1 = 2 ( x 2 ) = . 3 / . 4 = 3 / 4 ( x 3 ) = . 3 / . 1 = 3 ( x 4 ) = . 2 / . 4 = 1 / 2 b) ( x 4 ) < ( x 2 ) < ( x 1 ) < ( x 3 ). P ( reject H | H ) = P (Type I error) = 0 . 2 if the rejection region is { x 4 } . P ( reject H | H ) = P (Type I error) = 0 . 5 if the rejection region is { x 4 , x 2 } . c) This part of the exercise refers to the Bayesian paradigm. In the Bayesian paradigm, the competing hypotheses are assigned probabilitiesthe so-called prior probabili- ties before testing. After the test these probabilities are revised using the likelihood ratio (as we illustrate in the next part) to give the so-called posterior probabilities . The favored hypothesis is the hypothesis with the greater posterior probability. If the priors are equal, as in this part of this problem, the favored hypothesis is simply the more likely one. Thus, H is favored if X = x 1 or X = x 3 . d) In the Bayes paradigm, we assume that P ( H ) and P ( H 1 ) are positive numbers that add to 1. We then apply the following fact about conditional probability: P ( H i | A ) P ( A ) = P ( H i & A ) = P ( A | H i ) P ( H i ) ....
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This note was uploaded on 11/29/2011 for the course MATH 4056 taught by Professor Staff during the Fall '08 term at LSU.

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notes-11-29 - M4056 Take-Home Test Answers November 29,...

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