7 Engineering work on the railway network causes an increase in the journey time of commuters travelling into work each morning.

The increase in journey time, $T$ hours, is modelled by a continuous random variable with probability density function

$$\mathrm { f } ( t ) = \begin{cases} 4 t \left( 1 - t ^ { 2 } \right) & 0 \leqslant t \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$

(a) Show that $\mathrm { E } ( T ) = \frac { 8 } { 15 }$.\\
(b) (i) Find the cumulative distribution function, $\mathrm { F } ( t )$, for $0 \leqslant t \leqslant 1$.\\
(ii) Hence, or otherwise, for a commuter selected at random, find

$$\mathrm { P } ( \text { mean } < T < \text { median } )$$

\hfill \mbox{\textit{AQA S2  Q7}}