Question: Suppose that X1, . . . , Xn form a random sample from the normal distribution with unknown mean and unknown precision ,

Suppose that X1, . . . , Xn form a random sample from the normal distribution with unknown mean μ and unknown precision τ , and also that the joint prior distribution of μ and τ is the normal-gamma distribution satisfying the following conditions: E(τ) = 1, Var(τ ) = 1/3, Pr(μ > 3) = 0.5, and Pr(μ > 0.12) = 0.9. Determine the prior hyperparameters μ0, λ0, α0, and β0.

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