7 The continuous random variable \(X\) has a uniform distribution over the interval \([ 0 , \theta ]\) so that the probability density function is given by $$f ( x ) = \begin{cases} \frac { 1 } { \theta } & 0 \leqslant x \leqslant \theta
0 & \text { otherwise } \end{cases}$$ where \(\theta\) is a positive constant. A sample of \(n\) independent observations of \(X\) is taken and the sample mean is denoted by \(\bar { X }\).

1. The estimator \(T _ { 1 }\) is defined by \(T _ { 1 } = 2 \bar { X }\). Show that \(T _ { 1 }\) is an unbiased estimator of \(\theta\). It is given that the probability density function of the largest value, \(U\), in the sample is $$g ( u ) = \begin{cases} \frac { n u ^ { n - 1 } } { \theta ^ { n } } & 0 \leqslant u \leqslant \theta
   0 & \text { otherwise } \end{cases}$$
2. Find \(\mathrm { E } ( U )\) and show that \(\operatorname { Var } ( U ) = \frac { n \theta ^ { 2 } } { ( n + 1 ) ^ { 2 } ( n + 2 ) }\).
3. The estimator \(T _ { 2 }\) is defined by \(T _ { 2 } = \frac { n + 1 } { n } U\). Given that \(T _ { 2 }\) is also an unbiased estimator of \(\theta\), show that \(T _ { 2 }\) is a more efficient estimator than \(T _ { 1 }\) for \(n > 1\).