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 \(Y_{+}(x)\).

- \(P \%=\left(1-\alpha_{1}ight) 100 \%\) credible interval \(I_{\alpha_{1}}\) for the regressions line \(y(x)\).

- \(Q \%=\left(1-\alpha_{2}ight) 100 \%\) predictive interval \(I_{\alpha_{2}}^{+}\)for the next observation \(Y_{+}(x)\).

(a) Data: \(\{(33.74,260.1), \quad(28.71,226.7), \quad(39.9,300.3), \quad(43.29,321)\), \((14.2,112.3)\}\) Uncertainty: \(\sigma_{0}=4\) and \(n_{0}=5 . \alpha_{1}=0.05, \alpha_{2}=0.1\).

(b) Data: \(\{(1,2),(2,4),(3,3)\} . \sigma\) unknown. \(P \%=Q \%=90 \%\).

(c) Data: \(\{(2,10),(3,8),(4,8),(6,7)\} . \sigma_{0}=0.5\) and \(n_{0}=4 . \alpha_{1}=\alpha_{2}=\) 0.05 .

(d) Data: \(\{(0,0),(1,2),(2,7),(3,5)\} . \sigma\) unknown. \(P \%=90 \%, Q \%=95 \%\).

(e) Data: \(\{(-2,7),(1,5),(8,6)\} . \sigma\) unknown. \(\alpha_{1}=\alpha_{2}=0.02\).

(f) Data: \(\{(28,24),(66,69),(44,48),(39,44),(9,9),(1,15),(73,64)\), \((41,44)\} . \sigma_{0}=7, n_{0}=10 . P \%=Q \%=80 \%\).

(g) Data: \(\{(-2.9,0.8),(43.7,36.9),(16.4,11.7),(47.3,41.9),(25.5,16.9)\), \((22.1,23.1),(38.4,42.4),(35.3,38.8)\} . \sigma_{0}=5.5, n_{0}=6 . \alpha_{1}=\alpha_{2}=\) 0.02 .

(h) Data: \(\{(135.1,8.9),(208.7,16),.(241.4,16.5),(217.1,6.6),(215.3,7.7)\), \((154.8,2.4),(263.8,18.6)\} . \sigma\) unknown. \(P \%=95 \%, Q \%=90 \%\).