7 The waiting time, \(X\) minutes, for fans to gain entrance to see an event may be modelled by a continuous random variable having the distribution function defined by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 0
\frac { 1 } { 2 } x & 0 \leqslant x \leqslant 1
\frac { 1 } { 54 } \left( x ^ { 3 } - 12 x ^ { 2 } + 48 x - 10 \right) & 1 \leqslant x \leqslant 4
1 & x > 4 \end{cases}$$

1. 1. Sketch the graph of F.
   2. Explain why the value of \(q _ { 1 }\), the lower quartile of \(X\), is \(\frac { 1 } { 2 }\).
   3. Show that the upper quartile, \(q _ { 3 }\), satisfies \(1.6 < q _ { 3 } < 1.7\).
2. The probability density function of \(X\) is defined by $$\mathrm { f } ( x ) = \begin{cases} \alpha & 0 \leqslant x \leqslant 1
   \beta ( x - 4 ) ^ { 2 } & 1 \leqslant x \leqslant 4
   0 & \text { otherwise } \end{cases}$$
   1. Show that the exact values of \(\alpha\) and \(\beta\) are \(\frac { 1 } { 2 }\) and \(\frac { 1 } { 18 }\) respectively.
   2. Hence calculate \(\mathrm { E } ( X )\).