We observe a sample of 30 observations from a normal distribution with mean $\mu$ and precision $\tau$ . The data, $y_{1},\dotsc,y_{30}$ , are such that  

$$
\sum_{i=1}^{30}y_{i}=672\qquad\qquad\qquad{\mathrm{and}}\qquad\qquad\sum_{i=1}^{30}y_{i}^{2}=16193.
$$  

(a) Suppose that the value of $\tau$ is known to be 0.04 and that our prior distribution for $\mu$ is normal with mean 20 and variance 100. Find the posterior distribution of $\mu$ and evaluate a posterior 95% hpd interval for $\mu$ .  

(b) Suppose that we have a gamma $,(1,10)$ prior distribution for $\tau$ and our conditional prior distribution for $\mu$ given $\tau$ is normal with mean 20 and variance $(0.1\tau)^{-1}$ . Find the marginal posterior distribution for $\tau$ , the marginal posterior distribution for $\mu$ and the marginal posterior 95% hpd interval for $\mu$ .