Problem 2. (25 points) Consider a binary classification problem in which we want to determine the optimal decision surface. A point x is on the decision surface if P(Y = 1|x) = P(Y = 0|x). a) (10 points) Find the optimal decision surface assuming that each class-conditional distribution is defined as a two-dimensional Gaussian distribution: p ( x Y = k) =- (27 ) d/ 2 1 1/2 e 2 (x-mk) ER (x-mk) where k E {0, 1}, mo = (1, 2), m1 = (6,3), Zo = E1 = 12, P(Y = 0) = P(Y = 1) = 1/2, Id is the d-dimensional identity matrix, and |Ex| is the determinant of Ek. b) (5 points) Generalize the solution from part (a) using mo = (mol, moz), m1 = (m11, m12), Zo = El = o212 and P(Y = 0) # P(Y = 1). c) (10 points) Generalize the solution from part (b) to arbitrary covariance matrices Zo and E1. Discuss the shape of the optimal decision surface