4. The lifetime, \(X\), in tens of hours, of a battery has a cumulative distribution function \(\mathrm { F } ( x )\) given by
$$\mathrm { F } ( x ) = \left\{ \begin{array} { c c }
0 & x < 1
\frac { 4 } { 9 } \left( x ^ { 2 } + 2 x - 3 \right) & 1 \leqslant x \leqslant 1.5
1 & x > 1.5
\end{array} \right.$$
- Find the median of \(X\), giving your answer to 3 significant figures.
- Find, in full, the probability density function of the random variable \(X\).
- Find \(\mathrm { P } ( X \geqslant 1.2 )\)
A camping lantern runs on 4 batteries, all of which must be working. Four new batteries are put into the lantern.
- Find the probability that the lantern will still be working after 12 hours.