Consider a two$-$class classification problem with the following prior probabilities and class$-$conditional densities:

$*$P$($w$1)=0\mathrm{.}4,$ P$($w$2)=0\mathrm{.}6$

$*$P$($x$|$w$1)=$ N$($x; $2,1)$

$*$p$($x$|$w$2)=$ N$($x; $4,2)$

$($a$)$ Derive the decision boundary for minimum$-$error$-$rate classification.

$($b$)$ Calculate the misclassification error.

$($c$)$Discuss how the decision boundary and error would change if the priors were equal.