3 Linear Regression [2 pts] A closed form solution (normal equation) for linear regression problem is as follows: e = (XX)-1XTY. A probabilistic interpretation of linear regression tells us that we are relying on an assumption that each data point is actually sampled from a linear hyperplane, with some noise. The noise follows a zero-mean Gaussian distribution with constant variance. Specifically, Y = oxite (2) 2 where e" ~ N(0, o'[), 0 ER, and {X", Y'} is the i-th data point. In other words, we are assuming that each point is independent to each other and that every data point has the same variance. (a) Using the normal equation, and the model (Eqn. 2), derive the expectation E[0]. Note that here X is fixed, and only Y is random, i.e. "fixed design" as in statistics. [0.4 pts] (b) Similarly, derive the variance Var. [0.4 pts] (c) Under the white noise assumption above, someone claims that o follows Gaussian distribu- tion with mean and variance in (a) and (b), respectively. Do you agree with this claim? Why or why not? [0.5 pts] 3