7. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by $$f ( x ) = \left\{ \begin{aligned} \frac { 1 } { 20 } x ^ { 3 } , & 1 \leq x \leq 3
0 , & \text { otherwise } \end{aligned} \right.$$

1. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
2. Calculate \(\mathrm { E } ( X )\).
3. Show that the standard deviation of \(X\) is 0.459 to 3 decimal places.
4. Show that for \(1 \leq x \leq 3 , \mathrm { P } ( X \leq x )\) is given by \(\frac { 1 } { 80 } \left( x ^ { 4 } - 1 \right)\) and specify fully the cumulative distribution function of \(X\).
5. Find the interquartile range for the random variable \(X\). Some statisticians use the following formula to estimate the interquartile range: $$\text { interquartile range } = \frac { 4 } { 3 } \times \text { standard deviation. }$$
6. Use this formula to estimate the interquartile range in this case, and comment.