ORIE6700 Fall 2010 Homework 1 1. The Beta density for x ∈ [0 , 1] is given by f ( x | α, β ) = Γ( α + β ) Γ( α )Γ( β ) x α-1 (1-x ) β-1 where Γ is the gamma function and α, β > 0 are the parameters. Write this density in exponential family form (which proves that it is a member of this family). What is the natural parameter vector? What is T ( x )? How about the normalization factor ζ ( η )? 2. Derive the moment generating function of the chi-squared distribution, which has density f ( x | ν ) = 2-ν/ 2 Γ( ν/ 2) x ν/ 2-1 e-x/ 2 for ν > 0 and x > 0. Use the moment generating function to obtain an expression for the expectation of X ∼ f ( x | ν ). X 1 , . . . , X n be independent observations from a Poisson( θ ) distribution where θ > 0. (a) Show directly that ∑ n i =1 X i is suﬃcient for θ (b) Establish the same result using the factorization theorem.
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