6. (5 points) (i) If Y e {A, B}, show that assigning either outcome of Y as 1 and the other out- come as 0 would generate the same prediction result based on Y > 0.5, where Y is the least squares prediction of Y based on X whose first element is 1. (ii) If K = 2 and *1 = 12 = 1/2, then the LDA is equivalent to classify a to Class 1 if 6 (x) = (x - 4) E-1 (14 - /2) 2 0, where u = 2172 is the mean of X in the combined population of Y = 1 and Y = 2. (iii*) In the setup of (ii), suppose we label Class 1 as Y = 1 and Class 2 as Y = -1, and the observations frog and {yog are normalized to have mean 0. Show that the LDA and the least squares share the same classification rule. (iv* ) If K = 2 and Ye {0, 1}, show that arg ming(ro) E[I (Y # 9 (X)) |X = To] = 1, if Pr (Y = 1X = x0) > 0.5, Hint: note that 0, otherwise. I(Y # 9 (X)) = |Y - 9 (X)| 6 {0, 1}