Suppose the true average growth m of one type of plant during a 1year period is identical to that of a second type, but the variance of growth for the first type is σ2, whereas for the second type the variance is 4σ2. Let X1, ... , Xm be m independent growth observations on the first type [so E(Xi) = µ, V(Xi) = σ2], and let Y1, ... , Yn be n independent growth observations on the second type [E(Yi) = µ, V(Yi) = 4σ2].

a. Show that the estimator µ^ = δ+ (1 - δ)  is unbiased for µ (for 0 < δ < 1, the estimator is a weighted average of the two individual sample means).

b. For fixed m and n, compute V(µ^), and then find the value of d that minimizes V(µ^).