In the problems below, you are given observational data \(\left\{\left(x_{i}, y_{i}ight)ight\}\), and information about \(\sigma\). Find the ...

- posterior distribution of \(\tau\).

- posterior distribution of \(y(x)\).

- posterior predictive distribution of \(Y_{+}(x)\).

(a) Data: \(\{(77,3),(94,19),(87,8),(86,21),(70,1),(75,8),(81,1),(80,16)\), \((91,11),(98,22),(82,4),(86,21),(101,15),(75,2),(83,18),(87,7)\), \((78,15),(93,12),(77,8),(80,12)\} . \sigma\) unknown.

(b) Data: \(\{(23,-80),(2,2),(22,-79),(20,-67),(10,-37),(-8,41)\}\). \(\sigma_{0}=3\) and \(n_{0}=100000\).

(c) Data: \(\{(46,50),(100,105),(64,68),(3,8),(82,88),(48,57)\} . \sigma_{0}=2\) and \(n_{0}=1000\).