Following the previous question, prove that the most precise unbiased estimate is obtained by setting w₁ = w₂ = . Minimise V(w₁x₁ + w₂x₂) with respect to w₁ after substituting w₂ = 1 − w₁. You will need a knowledge of calculus to solve this.

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A random sample of two observations, x₁ and x₂, is drawn from a population. Prove that w₁x₁ + w₂x₂ gives an unbiased estimate of the population mean as long as w₁ + w₂ = 1. Prove that E(w₁x₁ + w₂x₂) = μ.