- The lifetime of a particular battery, \(T\) hours, is modelled using the cumulative distribution function
$$\mathrm { F } ( t ) = \left\{ \begin{array} { l r }
0 & t < 8
\frac { 1 } { 96 } \left( 74 t - \frac { 5 } { 2 } t ^ { 2 } + k \right) & 8 \leqslant t \leqslant 12
1 & t > 12
\end{array} \right.$$
- Show that \(k = - 432\)
- Find the probability density function of \(T\), for all values of \(t\).
- Write down the mode of \(T\).
- Find the median of \(T\).
- Find the probability that a randomly selected battery has a lifetime of less than 9 hours.
A battery is selected at random. Given that its lifetime is at least 9 hours,
- find the probability that its lifetime is no more than 11 hours.