1. [Derivation/ Conceptual] Consider the multiple linear regression model \" = X E + E' with the usual setup, where Y is the n x 1 column vector of responses, X is the n x (p + 1) matrix for the predictors (with intercept) and E' ~ M VN (0 021\"\"). Recall that we have the estimator (XTX) 1XTY and we dened the \"hat\" matrix H: X (XTX ) lXT. o Viewed as random vectors the tted values are a: X B and the residuals are E Y p. (a) State the distributions of" [.L and e as seen in class, making use of the hat matrix H (proof not required). (b) Dene the random vector E7 = [fl 6F, i.e., the 271 X 1 column vector obtained by concatenating the elements of [i and (5'. Compute the mean vector E(U), and covariance matrix Var(U). Hint: This \"can be expressed in block or partition form, since we already have covariance matrices of 11' and E' in part (a). Recall that matrix operations are also valid in block / partition form when the dimensions permit matrix multiplication. For example, _ H i_ H37 demng A (1H) , thenAy (IH)Q' ((3) Making use of the result in (b) and any facts from class, what can we conclude about the random vectors [1' and E ? Justify your answer clearly. (d) Describe in words what we would expect to see, if we make a scatterplot of the pairs (ll, 61), . . . , (fin, en) computed from the data, that is, plotting the tted values on the maxis and the residuals on the y-axis. Hint: If we drew a regression line through these (.111, (31-) pairs, what might the line look like