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

$$\mathrm { F } ( x ) = \begin{cases} 0 & x < 0 \\ \frac { 1 } { 2 } x & 0 \leqslant x \leqslant 1 \\ \frac { 1 } { 54 } \left( x ^ { 3 } - 12 x ^ { 2 } + 48 x - 10 \right) & 1 \leqslant x \leqslant 4 \\ 1 & x > 4 \end{cases}$$
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Sketch the graph of F.
\item Explain why the value of $q _ { 1 }$, the lower quartile of $X$, is $\frac { 1 } { 2 }$.
\item Show that the upper quartile, $q _ { 3 }$, satisfies $1.6 < q _ { 3 } < 1.7$.
\end{enumerate}\item The probability density function of $X$ is defined by

$$\mathrm { f } ( x ) = \begin{cases} \alpha & 0 \leqslant x \leqslant 1 \\ \beta ( x - 4 ) ^ { 2 } & 1 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$
\begin{enumerate}[label=(\roman*)]
\item Show that the exact values of $\alpha$ and $\beta$ are $\frac { 1 } { 2 }$ and $\frac { 1 } { 18 }$ respectively.
\item Hence calculate $\mathrm { E } ( X )$.
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{AQA S2 2008 Q7 [19]}}