8. The random variable \(X\) has probability density function $$\begin{array} { l l } f ( x ) = 1 + \frac { 3 \lambda x } { 2 } & \text { for } - \frac { 1 } { 2 } \leqslant x \leqslant \frac { 1 } { 2 }
f ( x ) = 0 & \text { otherwise } \end{array}$$ where \(\lambda\) is an unknown parameter such that \(- 1 \leqslant \lambda \leqslant 1\).

1. 1. Find \(\mathrm { E } ( X )\) in terms of \(\lambda\).
   2. Show that \(\operatorname { Var } ( X ) = \frac { 16 - 3 \lambda ^ { 2 } } { 192 }\).
2. Show that \(\mathrm { P } ( X > 0 ) = \frac { 8 + 3 \lambda } { 16 }\). In order to estimate \(\lambda , n\) independent observations of \(X\) are made. The number of positive observations obtained is denoted by \(Y\) and the sample mean is denoted by \(\bar { X }\).
3. 1. Identify the distribution of \(Y\).
   2. Show that \(T _ { 1 }\) is an unbiased estimator for \(\lambda\), where $$T _ { 1 } = \frac { 16 Y } { 3 n } - \frac { 8 } { 3 }$$
4. 1. Show that \(\operatorname { Var } \left( T _ { 1 } \right) = \frac { 64 - 9 \lambda ^ { 2 } } { 9 n }\).
   2. Given that \(T _ { 2 }\) is also an unbiased estimator for \(\lambda\), where $$T _ { 2 } = 8 \bar { X }$$ find an expression for \(\operatorname { Var } \left( T _ { 2 } \right)\) in terms of \(\lambda\) and \(n\).
   3. Hence, giving a reason, determine which is the better estimator, \(T _ { 1 }\) or \(T _ { 2 }\). \section*{END OF PAPER}