Let Y1, Y2, and Y3 be a random sample from a normal distribution with mean μ and variance σ^2, where both μ and σ^2 are unknown. Consider the following estimators for μ:
μ̂1 = (1/4)Y1 + (1/2)Y2 + (1/4)Y3
μ̂2 = (1/3)Y1 + (1/3)Y2 + (1/3)Y3
a) Show that μ̂1 and μ̂2 are unbiased estimators for μ.
b) Find the variances of μ̂1 and μ̂2.
c) Find the efficiency of μ̂2 relative to μ̂1, and determine which estimator is more efficient for μ.