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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 Θ ≡ “Norway”. Problem 4.1 By Taylor series expansion, we have the identity − log(1 − θ ) = ∑ ∞ x =1 θ x x , which is valid for all θ ∈ (0 , 1). From this fact, the quantity p ( x ; θ ) = θ x − x log(1 − θ ) , x = 1 , 2 , . . . defines a valid probability mass function for all θ ∈ (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 Γ( λ, p ) (b) the beta family of distribution functions β ( λ, p ) (c) the family defined on the parameters β 1 and β 2 by the linear regression: y i = β 1 + β 2 · x i + ε with ε ∼ N (0 , σ 2 ) and σ 2 known....
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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 Berkeley.
- Fall '08