If X 1 is the mean of a random sample of size n from a normal population with the mean and the variance

If X̅₁ is the mean of a random sample of size n from a normal population with the mean µ and the variance σ²₁, X̅₂ is the mean of a random sample of size n from a normal population with the mean µ and the variance σ²₂, and the two samples are independent, show that

(a) ω ∙ X̅₁ +(1 – ω) ∙ X̅₂, where 0 ≤ ω ≤ 1, is an unbiased estimator of µ;

(b) The variance of this estimator is a minimum when