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Suppose that $X_1,X_2,\ldots,X_n$ form a random sample from a Normal distribution with unknown mean $\mu$ and known variance $\sigma^2$, and the prior distribution of $\mu$ is a normal distribution with mean 0 and variance $\sigma^2$.

(a) Obtain the posterior distribution for $\sigma^2$.

(b) Show that if n is large then the posterior distribution of mu given that $X_i = x_i$, $(i = 1; ... ; n)$ will be approximately a normal distribution with mean $x_n$ and variance $\sigma^2/n$.

Edit: I believe this question has to do with conjugate priors so feel free to steer me in the right direction if this is incorrect! Thanks beforehand.

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We have $$\text{posterior} \propto \text{likelihood} \times \text{prior},$$ and we know the likelihood and prior are both Gaussian [density functions]. Therefore, the posterior also follows a Gaussian density function. (Note that this is NOT the same as saying that the product of Gaussian random variables is Gaussian, see this.)

Part (a) can be done by writing out the density functions and multiplying them. Most of it is done here (section 2.1-2.3).

For part (b), if you look at the expressions for the mean and variance of the posterior that you found in part (a), you will see that taking $n \to \infty$ gives you what you need.

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angryavian
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