6. Emily is monitoring the level of pollution in a river. Over a period of time she has found that the amount of pollution, \(X\), in a 100 ml sample of river water has a continuous distribution with probability density function \(\mathrm { f } ( x )\) given by $$f ( x ) = \left\{ \begin{array} { c c } \frac { 2 x } { a ^ { 2 } } & 0 \leqslant x \leqslant a
0 & \text { otherwise } \end{array} \right.$$ where \(a\) is a constant. Emily takes a random sample \(X _ { 1 } , X _ { 2 } , X _ { 3 } , \ldots , X _ { n }\) to try to estimate the value of \(a\).

1. Show that \(\mathrm { E } ( \bar { X } ) = \frac { 2 a } { 3 }\) and \(\operatorname { Var } ( \bar { X } ) = \frac { a ^ { 2 } } { 18 n }\) The random variable \(S = p \bar { X }\), where \(p\) is a constant, is an unbiased estimator of \(a\).
2. Write down the value of \(p\) and find \(\operatorname { Var } ( S )\). Felix suggests using the statistic \(M = \max \left\{ X _ { 1 } , X _ { 2 } , X _ { 3 } , \ldots , X _ { n } \right\}\) as an estimator of \(a\).
   He calculates \(\mathrm { E } ( M ) = \frac { 2 n } { 2 n + 1 } a\) and \(\operatorname { Var } ( M ) = \frac { n } { ( n + 1 ) ( 2 n + 1 ) ^ { 2 } } a ^ { 2 }\)
3. State, giving your reasons, whether or not \(M\) is a consistent estimator of \(a\). The random variable \(T = q M\), where \(q\) is a constant, is an unbiased estimator of \(a\).
4. Write down, in terms of \(n\), the value of \(q\) and find \(\operatorname { Var } ( T )\).
5. State, giving your reasons, which of \(S\) or \(T\) you would recommend Emily use as an estimator of \(a\). Emily took a sample of 5 values of \(X\) and obtained the following:
   5.3
   4.3
   \(\begin{array} { l l } 5.7 & 7.8 \end{array}\)
   6.9
6. Calculate the estimate of \(a\) using your recommended estimator from part (e).
7. Find the standard error of your estimate, giving your answer to 2 decimal places.