6 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} 0 & x < a ,
\mathrm { e } ^ { - ( x - a ) } & x \geqslant a , \end{cases}$$ where \(a\) is a constant. \(X _ { 1 } , X _ { 2 } , \ldots , X _ { n }\) are \(n\) independent observations of \(X\), where \(n \geqslant 4\).

1. Show that \(\mathrm { E } ( X ) = a + 1\).
   \(T _ { 1 }\) and \(T _ { 2 }\) are proposed estimators of \(a\), where $$T _ { 1 } = X _ { 1 } + 2 X _ { 2 } - X _ { 3 } - X _ { 4 } - 1 \quad \text { and } \quad T _ { 2 } = \frac { X _ { 1 } + X _ { 2 } } { 4 } + \frac { X _ { 3 } + X _ { 4 } + \ldots + X _ { n } } { 2 ( n - 2 ) } - 1 .$$
2. Show that \(T _ { 1 }\) and \(T _ { 2 }\) are unbiased estimators of \(a\).
3. Determine which is the more efficient estimator.
4. Suggest another unbiased estimator of \(a\) using all of the \(n\) observations.