Let X,..., Xniid N(, T) for u known and 7 unknown. We will use the conjugate...

  

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Let X₁,..., Xniid N(μ, T) for u known and 7 unknown. We will use the conjugate prior distri- bution T InvG(a, 3). (a) Find the Bayes estimator for 7 under the loss function L(0, a) = (b) Show that (given 7), E{E1(X; -μ)2} = 7. (Hint: first find the distribution of (X; -μ)².) (c) Show that this Bayes estimator can be written as a weighted average between Σ₁₁ (X₁-µ)², the estimator from (b), and a = [E{-¹}]¹. Let X₁,..., Xniid N(μ, T) for u known and 7 unknown. We will use the conjugate prior distri- bution T InvG(a, 3). (a) Find the Bayes estimator for 7 under the loss function L(0, a) = (b) Show that (given 7), E{E1(X; -μ)2} = 7. (Hint: first find the distribution of (X; -μ)².) (c) Show that this Bayes estimator can be written as a weighted average between Σ₁₁ (X₁-µ)², the estimator from (b), and a = [E{-¹}]¹.