6. A random sample of three independent variables \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) is taken from a distribution with mean \(\mu\) and variance \(\sigma ^ { 2 }\).
- Show that \(\frac { 2 } { 3 } X _ { 1 } - \frac { 1 } { 2 } X _ { 2 } + \frac { 5 } { 6 } X _ { 3 }\) is an unbiased estimator for \(\mu\).
An unbiased estimator for \(\mu\) is given by \(\hat { \mu } = a X _ { 1 } + b X _ { 2 }\) where \(a\) and \(b\) are constants.
- Show that \(\operatorname { Var } ( \hat { \mu } ) = \left( 2 a ^ { 2 } - 2 a + 1 \right) \sigma ^ { 2 }\).
- Hence determine the value of \(a\) and the value of \(b\) for which \(\hat { \mu }\) has minimum variance.