$(20$ points$).$ The losses of actions $a_1,a_2,$ and $a_3,$ depending on ${}_1,{}_2,$ and ${}_3$ are given in the

following table:

An observation $x$ has the following distribution depending on $$ :

$($a$)$ Pick three decision rules and evaluate their risks.

$($b$)$ Determine the minimax rule among the decision rules you listed in part $($a$).$

$($c$)$ Assuming a uniform prior on $(p({}_1)=p({}_2)=p({}_3)=\frac{1}{3}),$ determine the Bayes rule.

$($d$)$ Suppose that you observe $x=30.$ Use posterior analysis to determine the Bayes rule

among the decision rules you listed in part $($a$).$