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

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