5..8.4. What is the squared-error Loss Bayes estimate for the parameter e in a binomial pdf,
where 8 has a uniform distribution-that is, a noninformative prior? (Recall that a
uniform prior is a beta pdf for which r = s = 1).
5..8..S. In Questions 5.8.2-5.8.4, is the Bayes e:.timate u nbiased? ls i t asymptotical ly u nbiased?
5.8.6. Suppose that Y is a gamma random variable with parameters r and e and the prior is also
gamma with parameters s and µ. Show tha t the posterior pdf is gamma wi th parameters
r + s and y + µ.
5.8.7. Let Y1, Y2, ...• Y,, be a random sample from a gamma pdf with parameters ,. and e. where the prior distribution assigned to e is the gamma pdf with parameters s and JL Let W = Y1 + Y2 + ··· + Y,,. Find the posterior pdf for e.
5.8.8. Find the squared-error loss Bayes estimate for 8 in Question 5.8.7.
5Jl9. Consider, again, lhe scenario described in Example 5.8.2-a binomial random variable
X has parameters n and e, where the latter has a beta prior with integer parameters r
and s. Integrate the joint pdf Px (k I B)fe(fJ) with respect to e to show that lhe marginal
pdf of X is given by
k = 0, 1, ..., 11
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