Correction to HW9

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There is a typo in HW9, Question 4. It should read as follows:

Let X_1,\ldots,X_n be a random sample from a N(μ,σ2) distribution. Let τ = σ − 2, so we can write the distribution as N(μ,τ − 1).

Now suppose the prior distribution of (μ,τ) is

f(\mu,\tau) \propto \frac{1}{\tau}.

Show that the posterior density of μ,τ is equal to

f(\mu,\tau|x_1,\ldots,x_n) = f(\mu| \tau, x_1,\ldots,x_n) f(\tau | x_1,\ldots,x_n)

where f(\mu | \tau, x_1,\ldots,x_n) is a normal density with mean \bar{x} and variance 1 / (τn) and f(\tau|x_1,\ldots,x_n) is a gamma density with parameters

\frac{n-1}{2}

and

\left[ \sum_{i=1}^n (x_i-\bar{x})^2/2 \right]^{-1}.

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