7. Each day on the way to work, a commuter encounters a similar traffic jam. The length of time, in 10-minute units, spent waiting in the traffic jam is modelled by the random variable \(T\) with the cumulative distribution function:
$$\begin{array} { l l }
\mathrm { F } ( t ) = 0 & t < 0 ,
\mathrm {~F} ( t ) = \frac { t ^ { 2 } \left( 3 t ^ { 2 } - 16 t + 24 \right) } { 16 } & 0 \leq t \leq 2 ,
\mathrm {~F} ( t ) = 1 & t > 2 .
\end{array}$$
- Show that 0.77 is approximately the median value of \(T\).
- Given that he has already waited for 12 minutes, find the probability that he will have to wait another 3 minutes.
- Find, and sketch, the probability density function of \(T\).
- Hence find the modal value of \(T\).
- Comment on the validity of this model.