2 A particular species of reed that grows up to 2 metres in length is used for thatching. The lengths in metres of the reeds when harvested are modelled by the random variable \(X\) which has the following probability density function, \(\mathrm { f } ( x )\). $$f ( x ) = \begin{cases} \frac { 3 } { 16 } \left( 4 x - x ^ { 2 } \right) & \text { for } 0 \leqslant x \leqslant 2
0 & \text { elsewhere } \end{cases}$$

1. Sketch \(\mathrm { f } ( x )\).
2. Show that \(\mathrm { E } ( X ) = \frac { 5 } { 4 }\) and find the standard deviation of the lengths of the harvested reeds.
3. Find the standard error of the mean length for a random sample of 100 reeds. Once the harvested reeds have been collected, any that are shorter than 1 metre are discarded.
4. Find the proportion of reeds that should be discarded according to the model.
5. Reeds are harvested from a large area which is divided into several reed beds. A sample of the harvested reeds is required for quality control. How might the method of cluster sampling be used to obtain it?