Suppose can be defined by the density functions f() according to the values of a real parameter , where a b. The decisions are D = {d : a d b}, representing guesses as to the true value of . The loss function is L(, d) = | d| r , where r is a given positive value. The prior density on is (). Assume all ()'s or f's are positive throughout the sample space S. (a) (2pts) Show that if (|x) is the posterior density function of given that X = x, then a Bayes procedure is obtained by choosing (x) = d 0 to minimize R b a | d 0 | r(|x)d. [It is OK to simply quote our in-class discussions of how to compute Bayes procedures. In any event, do not try to find a formula for the minimizing d 0 in this part (a).] (b) (4pts) In particular, for "squared error loss" (r = 2), show that from (a) that a Bayes procedure is (x) = mean of posterior law (|x) of . (c) (4pts) For r = 1 ("absolute error loss"), show that a Bayes procedure is obtained as any median (not necessarily unique!) of the posterior law of . Since the crucial result from probability theory used in demonstrating this may be unfamiliar, part of this problem is to prove it: If g is a univariate probability density function with finite first moment, R |c|g()d is minimized if and only if c is a median of g.