$(10$ points$)$ Suppose we have features $xin{R}^p,$ a two$-$class response, with class sizes $N_1,N_2.$

Assume that the covariance matrix is independent of class; i$.$e$.,$ hat$({)}_k=$hat$()$ for $k=1,2.$ Show

that the LDA rule classifies to class $2$ if

$x^T$hat$({)}^{-1}(hat({)}_2-$hat$({)}_1)>\frac{1}{2}(hat({)}_2+$hat$({)}_1{)}^T$hat$({)}^{-1}(hat({)}_2-$hat$({)}_1)-log(\frac{N_2}{N_1}).$