Exercise 1.5.1 Let Y =(y1,y2,y3) be a random vector. Suppose that E(Y) ∈M, whereM is defined by M = {(a,a−b,2b)|a,b ∈ R}.

(a) Show thatM is a vector space.

(b) Find a basis forM.

(c) Write a linear model for this problem (i.e., find X such that Y = Xβ +e, E

(e) = 0).

(d) If β = (β1,β2) in part (c), find two vectors r = (r1, r2, r3) and s =

(s1, s2, s3) such that E(rY) = rXβ = β1 and E(sY) = β2. Find another vector t = (t1, t2, t3) with r = t but E(tY) =β1.