In the case of two groups, we might consider using linear regression as a classification method. Specifically, given data (91, x1), . .., (In, En) (where g1, . .., In each take one of two possible values 0 or 1), we choose Bo, / to minimize (gi - Bo - 2 7 B) 2 1= 1 We can then use the value of Bo + x 3 to classify an observation a; for example, we might classify ac to g = 1 if Bo + a3 is greater than some threshold. (a) Define n 1 1 Show that n (b) Now define n (1 - gi)Ci Co 1= 1 (1 - gi) 1= 1 n gi to be the means of observations within the two groups and S = 7 1 n - 21 1 to be an estimate of a common covariance matrix for the two groups. Show that 3 -KS-1(1 -x0) for some constant k. (Hint: Note that = 1=1 I'M : I'M3 = gi(xi - x1)(x -x1)+(1 -9:) (x; -xo)(x -XO) and x = Xx1 + (1 - X)xo where ) = (91 + . . + gn).)