2. Jeremy Mixture. (a) Show that for likelihood f(x|0) and mixture prior TT (0 ) = ETTI(0) + (1 - E) T2(0), 0 EO, the posterior is a mixture of T(0|x) = ('m(0|x) + (1 -E') 12(0/x), where f(x|0)T. (0) mi(I) mi(x) = f(x|0)7:(0)do, i = 1,2, and Emi(I) ' Em (x) + (1 -()m2(x) (b) Now we assume X|0 ~ N(0, 80) and the prior for 0 is a mixture 0 ~ T (0) = =N(110, 60) + =N(100, 200). Find the posterior and Bayes estimator for 0 if X = 98