1. The length of time, \(T\), minutes, spent completing a particular task has probability density function

$$f ( t ) = \left\{ \begin{array} { c c } \frac { 1 } { 2 } ( t - 1 ) & 1 < t \leqslant 2
\frac { 1 } { 16 } \left( 14 t - 3 t ^ { 2 } - 8 \right) & 2 < t \leqslant 4
0 & \text { otherwise } \end{array} \right.$$

1. Use algebraic integration to find \(\mathrm { E } ( T )\) Given that \(\mathrm { E } \left( T ^ { 2 } \right) = \frac { 267 } { 40 }\)
2. find \(\operatorname { Var } ( T )\)
3. Find the cumulative distribution function \(\mathrm { F } ( t )\)
4. Find the 20th percentile of the time taken to complete the task.
5. Find the probability that the time spent completing the task is more than 1.5 minutes. Given that a person has already spent 1.5 minutes on the task,
6. find the probability that this person takes more than 3 minutes to complete the task.