In the case of normally distributed classes, discriminant functions are linear (straight lines, planes, and hyperplanes for two-, three-, and n-dimensional feature vectors, respectively) when the covariances matrices of corresponding classes are equal. Confirm this by deriving discriminant functions for a binary classification problem.

Given:

P(x ) q) = (27 ) /2 21/2 exp ( - 4 (x - M, ) ' E-' (x- 1, ) );q=1,2 Prove that linear discriminant functions: gq ( x ) = HE -x - HE Hq+ In P(vg); q = 1, 2 And decision boundary g(x) = g1(x) - g2(x) = 0 is given by: g(x) = w x+wo=0 wx + wo = (H[ - HJ ) E -' x - f ( Hi E- 'H, - MJ E-' H2) + In P(VI) P(V2)