A random sample of 2 observations, X_1 and X_2, is taken from a population with unknown mean $\mu$ and unknown variance $\sigma^2$.

(a) Explain why $\dfrac{X_1-X_2}{\sigma}$ is not a statistic.

(b) Show that $S$ is a biased estimator of $\mu$

(c) Hence find the bias, in terms of $\mu$, when $S$ is used as an estimator of $\mu$

Given that $Y=aX_1+bX_2$ is an unbiased estimator of $\mu$, where $a$ and $b$ are constants,

(d) find an equation, in terms of $a$ and $b$, that must be satisfied.

(e) Using your answer to part (d), show that $\mathrm{Var}(Y)=(2a^2-2a+1)\sigma^2$