- The time, in minutes, spent waiting for a call to a call centre to be answered is modelled by the random variable \(T\) with probability density function
$$f ( t ) = \left\{ \begin{array} { l c }
\frac { 1 } { 192 } \left( t ^ { 3 } - 48 t + 128 \right) & 0 \leqslant t \leqslant 4
0 & \text { otherwise }
\end{array} \right.$$
- Use algebraic integration to find, in minutes and seconds, the mean waiting time.
- Show that \(\mathrm { P } ( 1 < T < 3 ) = \frac { 7 } { 16 }\)
A supervisor randomly selects 256 calls to the call centre.
- Use a suitable approximation to find the probability that more than 125 of these calls take between 1 and 3 minutes to be answered.