2. Let X1, . . . ,X10 be a random sample from a population with mean and variance 2, and Y1, . . .

2. Let X1, . . . ,X10 be a random sample from a population with mean μ and variance σ2, and Y1, . . . ,Y10 be a random sample from another population with mean also equal to μ and variance 4σ2. The two samples are independent.

(a) Show that for any α, 0 ≤ α ≤ 1, !μ = αX +(1−α)Y is unbiased for μ.

(b) Obtain an expression for the MSE of !μ.

(c) Is the estimator X preferable over the estimator 0.5X + 0.5Y? Justify your answer.