Question show as below in the picture.

(gonsider jhe model Y, 2 0 + lXi | e, for 3' = 1, . . . , n, where E,- {if NU], 0'2), and let ,3 2 (g, ,81) be the least squares estimates of the coefcients (g, 51). (a) Derive the distribution of ,9 (hint: it should be a 2-d normal distribution. Why?) (b) Suppose we observe a new covariate Xn+1- Assuming 02 is known, use your answer above to derive a 99% condence interval for E(Y,,+1 | Xn+1)* (c) Now suppose o2 is unknown. If you simply replace 02 with {993ml}2 in the interval you derived in (b), do you think the new interval is too wide, too narrow, or valid? That is, if you repeat the experiment many times, do you think this new interval will contain E(Y,.,+1 | Xn+l) more than, less than, or approximately 99% of the time? Explain. (d) Now derive a 99% condence interval for Yn+1 assuming o2 is known. Is this interval larger or smaller than the one from part (b)? Does this agree with your intuition? Explain