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The Gamma distribution is given by
p(_ j_; _) =__􀀀(_)__􀀀1e􀀀__
 
1. Compute the expected value of the Gamma distribution. Show all work.
 
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2. Given a series of observations x1; : : : ; xn where xi _ N(_; _ ) for a known _, compute the posterior distribution of the precision _ , i.e. the inverse of the variance, when we place a Gamma distribution as the prior distribution on _ .
 
3. Derive an expression that shows that the posterior mean for _ can be written as a linear combination of the prior mean and the maximum likelihood estimator for _ . What happens as the number of points goes to in_nity?
 
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