Let X 1 and X 2 be independent, each with unknown mean and known variance

Let X₁ and X₂ be independent, each with unknown mean μ and known variance σ² = 1.

a. Let μ̂₁ = X₁ + X₂/2. Find the bias, variance, and mean squared error of μ̂₁.

b. Let μ̂₂ = X₁ + 2X₂/3. Find the bias, variance, and mean squared error of μ̂₂.

c. Let μ̂₃ = X₁ + X₂/4. Find the bias, variance, and mean squared error of μ̂₃.

d. For what values of μ does μ̂₃ have smaller mean squared error than μ̂₁?

e. For what values of μ does μ̂₃ have smaller mean squared error than μ̂₂?