5. The probability density function of the continuous random variable \(X\) is given by $$\begin{array} { l l } f ( x ) = \frac { 3 x ^ { 2 } } { \alpha ^ { 3 } } & \text { for } 0 \leqslant x \leqslant \alpha
f ( x ) = 0 & \text { otherwise. } \end{array}$$ \(\bar { X }\) is the mean of a random sample of \(n\) observations of \(X\).

1. 1. Show that \(U = \frac { 4 \bar { X } } { 3 }\) is an unbiased estimator for \(\alpha\).
   2. If \(\alpha\) is an integer, what is the smallest value of \(n\) that gives a rational value for the standard error of \(U\) ?
2. \(\quad \bar { X } _ { 1 }\) and \(\bar { X } _ { 2 }\) are the means of independent random samples of \(X\), each of size \(n\). The estimator \(V = 4 \bar { X } _ { 1 } - \frac { 8 } { 3 } \bar { X } _ { 2 }\) is also an unbiased estimator for \(\alpha\).
   1. Show that \(\frac { \operatorname { Var } ( U ) } { \operatorname { Var } ( V ) } = \frac { 1 } { 13 }\).
   2. Hence state, with a reason, which of \(U\) or \(V\) is the better estimator.