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Unformatted text preview: UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 5 Fall 2006 Issued: Thursday, September 14, 2006 Due: Thursday, September 21, 2006 Graded exercises Problem 5.1 a) We want to show that R ( , ) = E ( ( X- ) ) does not depend on . To do that write: E ( ( X- ) 2 ) = E ( X- ) 2 = = 1 where the second equality follows from the fact that the variance of a Poi( ) random variable has variance . b) From item c below, we have that for the Gamma( a,b ) prior a Bayes estimator is: ( X ) = a + i X i- 1 n + b By constructing a sequence of priors with b k 0 and a k 1, we have: k ( X ) = a k- 1 + i X i b k + n Hence: k ( X ) i X i n and the sequence of priors approaches the uniform distribution on R + . c) We have that: ( ) = exp[( a- 1)log( )- b + a log( b )- log ( a )] and p ( x | ) = exp "- n + X i x i log( ) # 1 Q i x i ! So: p ( | x ) exp " ( a + X i X i- 1)log( )- ( b + n ) # 1 which corresponds to a ( a + i X i ,b + n ) family (could also get this from being conjugate prior to Poisson).conjugate prior to Poisson)....
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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.
- Fall '08