A random sample of 2 observations, \(X _ { 1 }\) and \(X _ { 2 }\), is taken from a population with unknown mean \(\mu\) and unknown variance \(\sigma ^ { 2 }\)
Explain why \(\frac { X _ { 1 } - X _ { 2 } } { \sigma }\) is not a statistic.
Hence find the bias, in terms of \(\mu\), when \(S\) is used as an estimator of \(\mu\)
Given that \(Y = a X _ { 1 } + b X _ { 2 }\) is an unbiased estimator of \(\mu\), where \(a\) and \(b\) are constants,
find an equation, in terms of \(a\) and \(b\), that must be satisfied.
Using your answer to part (d), show that \(\operatorname { Var } ( Y ) = \left( 2 a ^ { 2 } - 2 a + 1 \right) \sigma ^ { 2 }\)