91Ó°ÊÓ

Predictive distributions: let \(y\) be the number of 6 's in 1000 rolls of a fair die. (a) Sketch the approximate distribution of \(y\), based on the normal approximation. (b) Using the normal distribution table, give approximate \(5 \%, 25 \%, 50 \%\), \(75 \%\), and \(95 \%\) points for the distribution of \(y\).

Normal distribution with unknown mean: a random sample of \(n\) students is drawn from a large population, and their weights are measured. The average weight of the \(n\) sampled students is \(\bar{y}=150\) pounds. Assume the weights in the population are normally distributed with unknown mean \(\theta\) and known standard deviation 20 pounds. Suppose your prior distribution for \(\theta\) is normal with mean 180 and standard deviation 40 . (a) Give your posterior distribution for \(\theta .\) (Your answer will be a function of \(n .\) ) (b) A new student is sampled at random from the same population and has a weight of \(\tilde{y}\) pounds. Give a posterior predictive distribution for \(\tilde{y}\). (Your answer will still be a function of \(n .\) ) (c) For \(n=10\), give a \(95 \%\) posterior interval for \(\theta\) and a \(95 \%\) posterior predictive interval for \(\hat{y}\). (d) Do the same for \(n=100\).

Posterior intervals: unlike the central posterior interval, the highest posterior interval is not invariant to transformation. For example, suppose that, given \(\sigma^{2}\), the quantity \(n v / \sigma^{2}\) is distributed as \(\chi_{n}^{2}\), and that \(\sigma\) has the (improper) noninformative prior density \(p(\sigma) \propto \sigma^{-1}, \sigma>0\) (a) Prove that the corresponding prior deusity for \(\sigma^{2}\) is \(p\left(\sigma^{2}\right) \propto \sigma^{-2}\). (b) Show that the \(95 \%\) highest posterior density region for \(\sigma^{2}\) is not the same as the region obtained by squaring a posterior interval for \(\sigma\).