I am having difficulty with these proofs. I heard from other students that the exams are mostly proofs so if you could provide explanations when proving these questions it would be very much appreciated, that way I can try and solve similar proofs.

Q.1) a) Represent the predicted value Y, as a linear function of the observations Y, Y,,., Y, and derive the mean and variance of Y, using the fact that for any constants a, , a,..., a, and independent observations Y,, Y2,..., Y, where Y, ~ N(H;, 6, ), E: Car, ; = [a,M, and V {Car) = Caro,. 1=1 1=1 b) Show the steps to conclude under normality of errors that Y - ( Bo + BIX . ) ~ ! ( 1 - 2 ) where ( 2) represent the student's t-random variable with (n-2) degrees of freedom. c) Use the formula for of {Y, } to derive the expression for o {boy