4. Light bulbs produced in a certain factory have lifetimes, in 100 s of hours, whose distribution is modelled by the random variable \(X\) with probability density function
$$\begin{array} { l l }
\mathrm { f } ( x ) = \frac { 2 x ( 3 - x ) } { 9 } , & 0 \leq x \leq 3
\mathrm { f } ( x ) = 0 & \text { otherwise }
\end{array}$$
- Sketch \(\mathrm { f } ( x )\).
- Write down the mean lifetime of a bulb.
- Show that ten times as many bulbs fail before 200 hours as survive beyond 250 hours.
- Given that a bulb lasts for 200 hours, find the probability that it will then last for at least another 50 hours.
- State, with a reason, whether you consider that the density function \(f\) is a realistic model for the lifetimes of light bulbs.
\section*{STATISTICS 2 (A) TEST PAPER 2 Page 2}