Consider X1, · · · , Xn|μ i.i.d ∼ N (μ, σ2), where σ2 is known. The prior distribution of μ is N (μ0, σ2 0 ), where μ0 and σ2 0 are known. (a) Show that ˆμBayes is just the prior mean with no data given but is the sample mean in large samples. (b) Your prior distribution for μ is N (10, 25). Compute the prior probability that μ is larger than 20. Hint: Use the normal table or R function pnorm. (c) Suppose you know V ar(Xi) = 5 and your sample of size 50 has the average value of 20. With the prior distribution in (b), compute the posterior probability that μ is larger than 20. Hint: Use the normal table or R function pnorm.

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