A random sample of two observations X1 and X2 is taken from a population with unknown mean $\mu$ and unknown variance $\sigma^2$

(a) Explain why $\dfrac{X_1-\mu}{\sigma}$ is not a statistic.

(b) Explain what you understand by an unbiased estimator for $\mu$

Two estimators for $\mu$ are U1 and U2 where

(c) Show that both U1 and U2 are unbiased estimators for $\mu$

The most efficient estimator among a group of unbiased estimators is the one with the smallest variance.

(d) By finding the variance of U1 and the variance of U2 state, giving a reason, the most efficient estimator for $\mu$ from these two estimators.