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Unformatted text preview: UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 4 Fall 2006 Issued: Thursday, September 21, 2006 Due: Thursday, September 28, 2006 Note: For this problem set, vector Norway. Problem 4.1 By Taylor series expansion, we have the identity log(1 ) = x =1 x x , which is valid for all (0 , 1). From this fact, the quantity p ( x ; ) = x x log(1 ) , x = 1 , 2 , . . . defines a valid probability mass function for all (0 , 1). Compute the mean and variance of a random variable X with this log series distribution . Problem 4.2 Find conjugate distributions for the following family of distributions: (a) the gamma family of distribution functions ( , p ) (b) the beta family of distribution functions ( , p ) (c) the family defined on the parameters 1 and 2 by the linear regression: y i = 1 + 2 x i + with N (0 , 2 ) and 2 known....
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