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Unformatted text preview: Second Edition 9-5 9.13 a. For Y =- (log X )- 1 , the pdf of Y is f Y ( y ) = y 2 e- /y , 0 < y < , and P ( Y/ 2 Y ) = Z 2 y 2 e- /y dy = e- /y 2 = e- 1 / 2- e- 1 = . 239 . b. Since f X ( x ) = x - 1 , 0 < x < 1, T = X is a good guess at a pivot, and it is since f T ( t ) = 1, < t < 1. Thus a pivotal interval is formed from P ( a < X < b ) = b- a and is : log b log x log a log x . Since X uniform(0 , 1), the interval will have confidence .239 as long as b- a = . 239. c. The interval in part a) is a special case of the one in part b). To find the best interval, we minimize log b- log a subject to b- a = 1- , or b = 1- + a . Thus we want to minimize log(1- + a )- log a = log ( 1+ 1- a ) , which is minimized by taking a as big as possible. Thus, take b = 1 and a = , and the best 1- pivotal interval is n : 0 log log x o . Thus the interval in part a) is nonoptimal. A shorter interval with confidence coefficient .239 isthe interval in part a) is nonoptimal....
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- Spring '12