Consider data Y1,...,Yn that is a random (i.e. independent) sample from a Normal distribution with unknown mean and known variance 2 You wish to infer from the data using a Bayesian approach and select a prior on that is Normal with mean 0 and variance 2 1. the likelihood of the data in the form of a probability density function, simplifying the expression as much as possible. Then write down the prior on in the form of a probability density function. 2.derive the posterior distribution for , then state the name and parameters of this distribution. Since you already know the final answer to this question (see notes), points will be awarded for the derivation only, so show all your work and do not skip steps.

3. the posterior mean you found in part b in terms of the posterior variance. Interpret the posterior mean in terms of how it captures information from the data versus information from the prior. Be sure to discuss what happens to the mean when the sample size n grows.