A variation of the model in Exercise 7.19 is to let the random variables Y1,..., Yn satisfy
Yi = βXi + εi, i = 1,... , n,
where X1,... , Xn are independent n(μ, τ2) random variables, εi,..., εn are iid n(0, σ2), and the Xs and εs are independent. Exact variance calculations become quite difficult, so we might resort to approximations. In terms of μ, τ2, and σ2, find approximate means and variances for
(a) ∑XiYi / ∑X2i.
(b) ∑Yi / ∑Xi.
(c) ∑(Yi / Xi)/n.