stat210a_2007_hw5_solutions - UC Berkeley Department of...

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Unformatted text preview: UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 5- Solutions Fall 2007 Issued: Thursday, September 27 Due: Thursday, October 4 Problem 5.1 (a) For the Gamma density, we have: p ( x | ) exp log( x )- px + log p ( ) I ( p 0) I ( 0) And hence, a conjugate family is an exponential family on ( ,p ) with natural statistics ,p, log p ( ) , that is: ( ,p ) = exp t 1 + t 2 p + t 3 log p ( ) - ( t ) I ( p 0) I ( 0) Ideally, we would now prove that the subset: T = { t : Z exp t 1 + t 2 p + t 3 log p ( ) d ( p, ) I ( p 0) I ( 0) d ( p, ) < } is nonempty and for t T ( t ): ( t ) = log Z exp t 1 + t 2 p + t 3 log p ( ) I ( p 0) I ( 0) d ( p, ) (b) For the Beta density, we have: p ( x | ) exp a log( x ) + b log( x ) + log ( a + b ) ( a )( b ) And hence, a conjugate family is an exponential family on ( a,b ) with natural statistics a,b, ( a + b ) ( a )( b ) , that is: ( a,b ) = exp t 1 a + t 2 b + t 3 log ( a + b ) ( a )( b ) - ( t ) I ( a 0) I ( b 0) 1 which can be normalized to be a density for: T = { t : Z exp t 1 a + t 2 b + t 3 log ( a + b ) ( a )( b ) I ( a 0) I ( b 0) d ( a,b ) < } by setting: ( t ) = log Z exp t 1 a + t 2 b + t 3 log ( a + b ) ( a )( b ) I ( a 0) I ( b 0) d ( a,b ) Notice that for t 1 < ,t 2 < 0 and t 3 = 0, this corresponds to the distribution of two independent random variables with exponential distributuons....
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This note was uploaded on 10/17/2009 for the course STAT 210a taught by Professor Staff during the Fall '08 term at University of California, Berkeley.

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stat210a_2007_hw5_solutions - UC Berkeley Department of...

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