7 The continuous random variable \(X\) has probability density function defined by $$f ( x ) = \begin{cases} \frac { 1 } { 5 } ( 2 x + 1 ) & 0 \leqslant x \leqslant 1
\frac { 1 } { 15 } ( 4 - x ) ^ { 2 } & 1 < x \leqslant 4
0 & \text { otherwise } \end{cases}$$

1. Sketch the graph of f.
2. 1. Show that the cumulative distribution function, \(\mathrm { F } ( x )\), for \(0 \leqslant x \leqslant 1\) is $$\mathrm { F } ( x ) = \frac { 1 } { 5 } x ( x + 1 )$$
   2. Hence write down the value of \(\mathrm { P } ( X \leqslant 1 )\).
   3. Find the value of \(x\) for which \(\mathrm { P } ( X \geqslant x ) = \frac { 17 } { 20 }\).
   4. Find the lower quartile of the distribution.