Bayes$-$optimal classifier. The optimal misclassification error is achieved by the Bayes optimal classifier. Derive the Bayes optimal classifier for Y as a function of r and $\backslash$sigma $.$ Bayes$-$optimal classifier. The optimal misclassification error is achieved by the Bayes optimal classifier. That is$,$ the Bayes optimal classifier predicts

$=$f$^*($x$)=$y in $\{0,1\}$max P$($Y$=$y $|$ X$=$x$).$

$1.$ Consider a scalar feature X in $^2$ and a binary random variable Y$,$ for which

P$($X $|$ Y$=0)=\{[1/\backslash$pi r$^2$ X$<=$ r; $0$ otherwise $].$

P$($X $|$ Y$=1)=([[0$; $0]],\backslash$sigma $^2[[10$; $01]])=1/2\backslash$pi $\backslash$sigma $^2$exp$(-$X$^2/2\backslash$sigma $^2),$ and

P$($Y$=0)=$c P$($Y$=1),$ with

where r$,\backslash$sigma $>0$ and $0$