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8. Suppose X ~ N(0, 1) and T(0) = 0.9To(0) + 0.1q(0), where no is N(0, 2) and q is N(0, 10). If r = 1 is observed, find the posterior distribution and the posterior mean. The posterior distribution [15 marks] T(ex) xx 0.9 . 1 = exp 02 - 20 + 1 02 ) 02 - 20+1 V2 92 2 + 0.1. - exp V10 2 20 1 = 0.9 . 302 - 40 + 2 exp 1 1102 - 200 + 10 V2 + 0.1 . = exp V10 20 0.9 . 1 exp 3(0 - 3)2 + 3 1 V2 + 0.1 . - V10 exp 11(0 - 19)2+ 20 0.9 . - 1 (0 - )2 V2 . exp(-1/6) exp (- 2 . 2/3 +0.1 . . exp(-1/22) exp (0 - 19)2 0.9 2 - 10/11 = V3 exp(-1/6) - - (0 - 2)2 0.1 V2/3 = . exp 2 - 2/3 V11 . exp(-1/22) . - (0 - 10)2 = exp V10/11 2 - 10/11 The posterior mean is [ 10 marks] exp(-1/6) + * VII exp(-1/22) ) . exp(-1/6) . 3 + Vli . exp(-1/22) . 11 = 0.68157