5102-Solution-Set-03

5102-Solution-Set-03 - SOLUTIONS FOR PROBLEM SET 3 STAT...

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Unformatted text preview: SOLUTIONS FOR PROBLEM SET 3 STAT 5102-004 Since the first three problems deal with binomial sampling and beta priors, a few preliminary remarks will help to set the context. Let Y be distributed binomially over { ,...,n } with success probability . The likelihood function for given y satisfies lik( | y ) y (1- ) n- y . A Beta( , ) prior for has mean = + . Its density satisfies ( ) - 1 (1- ) - 1 . The posterior density is proportional to the prior density times the likelihood ( | y ) + y- 1 (1- ) + n- y- 1 . Since the right hand side of this proportionality is the the kernel of a beta density, the posterior distribution is a Beta( + y, + n- y ). (1) Let Y Bin( n, ). Assume a Beta(5 , 10) prior. The posterior is Beta(5+ y, 10+ n- y ). For n = 20 and y = 1, the posterior mean 5 + y 5 + 10 + n = 6 35 is the Bayes estimate of . (2) Let Y = n X n The Bayes estimator under squared error loss is the posterior mean n = + n X n + + n = n X n + + n + + + n = n + + n X n + + + + n . Let n = n + + n . The posterior mean n = n X n + (1- n ) is an average of the prior mean and the sample mean. Since + is fixed, the ratio of n to + + n goes to unity as n increases. Thus n- 1 and (1- n ) - 0 as n- ....
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5102-Solution-Set-03 - SOLUTIONS FOR PROBLEM SET 3 STAT...

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