HW2 - ORIE6700 Homework 2 Fall 2010 1 Consider an...

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ORIE6700 Fall 2010 Homework 2 1. Consider an experiment in which, for given θ , the outcome X has density f ( x | θ ) = 2 x/θ 2 , 0 < x < θ. Let π be a prior density for θ . (a) Find the posterior density of θ when π ( θ ) = 1 , 0 θ 1 . (b) Find the posterior density of θ when π ( θ ) = 3 θ 2 , 0 θ 1 . (c) Find E ( θ | x ) for the two priors given in (a), (b). (d) Now suppose X 1 , . . . , X n are iid with the same distribution as X . Find the posterior density of θ given X i = x i , i = 1 , . . . , n when π ( θ ) = 1 , 0 θ 1 . 2. Let f ( x | θ ) = exp {- ( x - θ ) } , 0 < θ < x, and let π ( θ ) = 2 exp {- 2 θ } , θ > 0 . Find the posterior density π ( θ | x ) . 3. Suppose the possible states of nature are θ 1 , θ 2 and the possible actions are a 1 , a 2 , a 3 , and the loss function ` ( θ, a ) is given in the following table: Suppose
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