Given x 2 Rn and y 2 f0, 1g, assume Pry = k = k > 0

Given x 2 Rn and y 2 f0, 1g, assume Pr¹y = kº = k > 0 for k = 0, 1 with ¹0 + 1 = 1º, and the conditional distribution of x given y is p¹x j yº = N¹x j y, yº, where 0, 1 2 Rn are two mean vectors (with 0 , 1), and 0, 1 2 Rdd are two covariance matrices.

a. What is the unconditional density of x (i.e., p¹xº)?

b. Assume that 0 = 1 =  is a positive definite matrix. Derive the MAP decision rule. What is the nature of the separation boundary between two classes? Show the procedure.

c. Assume that 0 , 1 are two positive-definite matrices. Derive the MAP decision rule. What is the nature of the separation boundary between two classes? Show the procedure.