6 The continuous random variable \(Z\) has probability density function
$$f ( z ) = \left\{ \begin{array} { c c }
\frac { 4 z ^ { 3 } } { k ^ { 4 } } & 0 \leqslant z \leqslant k
0 & \text { otherwise }
\end{array} \right.$$
where \(k\) is a parameter whose value is to be estimated.
- Show that \(\frac { 5 Z } { 4 }\) is an unbiased estimator of \(k\).
- Find the variance of \(\frac { 5 Z } { 4 }\).
The parameter \(k\) can also be estimated by making observations of a random variable \(X\) which has mean \(\frac { 1 } { 2 } k\) and variance \(\frac { 1 } { 12 } k ^ { 2 }\). Let \(Y = X _ { 1 } + X _ { 2 } + X _ { 3 }\) where \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) are independent observations of \(X\).
- \(c Y\) is also an unbiased estimator of \(k\). Find the value of \(c\).
- For the value of \(c\) found in part (iii), determine which of \(\frac { 5 Z } { 4 }\) and \(c Y\) is the more efficient estimator of \(k\).