$3.$ Consider a two$-$category classification problem with two$-$dimensional feature vector X $=($XI$,$ x$2).$ The two categories are o$,$ and $0.$ plcx$|)-1($a$)$:$-).$ plx$|$on$)-$ N$(?)$:$).$ P$($q$)=$ P$(0.)=\}$ and $($a$)$ Calculate the Bayes decision boundary. $($b$)$ Randomly draw $50$ patterns from each of the two class$-$conditional densities and plot them in the two$-$dimensional feature space. Also draw the decision boundary $($from $($a$))$ on this plot. $($c$)$ Calculate the Bhattacharya error bound. $($d$)$ Generate $1000$ additional test patterns from each class and determine the empirical error rate based on the decision boundary in $(3$a$).$ Compare the empirical error with the bound in part $(3$c$).$