Given a two class problem for which the feature vectors of the samples are in Normal

distribution, i$.$e$.,N({}_{{?}_{?}}(1),K_{{?}_{?}}(1))$ if $x_{i}n{}_1,$ and $N({}_2,K_{{?}_{?}}(2))$ if $x_{i}n{}_2$

Where $,{}_{{?}_{?}}(1)=[0,0{]}^T,{}_{{?}_{?}}(2)=[5,5{]}^T,K_{{?}_{?}}(1)=\left[\begin{array}{cc}2& 1\\ 1& 3\end{array}\right],$ and $,K_{{?}_{?}}(2)=\left[\begin{array}{cc}2& 1\\ 1& 3\end{array}\right]$

And $P({}_1)=0\mathrm{.}4,P({}_2)=0\mathrm{.}6$

$($a$)$ Apply Bayesian decision rule to find the discriminant function.

$($b$)$ Determine the classes of samples $x_{?}=[1,3{]}^T$ and $x_{?}=[3,3{]}^T.$

$($c$)$ Sketch the distributions and the decision regions.

$($d$)$ Find the discriminant function if $P({}_1)=P({}_2)=0\mathrm{.}5$

Given a $2$D matrix $K_{?}=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right],$ its inverse is $K_{{?}^{?}}(-1)=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -C\\ -b& a\end{array}\right].$