The continuous random variable \(L\) represents the error, in metres, made when a machine cuts poles to a target length. The distribution of \(L\) is a continuous uniform distribution over the interval \([ 0,0.5 ]\)
Find \(\mathrm { P } ( L < 0.4 )\).
Write down \(\mathrm { E } ( L )\).
Calculate \(\operatorname { Var } ( L )\).
A random sample of 30 poles cut by this machine is taken.
Find the probability that fewer than 4 poles have an error of more than 0.4 metres from the target length.
When a new machine cuts poles to a target length, the error, \(X\) metres, is modelled by the cumulative distribution function \(\mathrm { F } ( x )\) where
$$\mathrm { F } ( x ) = \left\{ \begin{array} { c c }
0 & x < 0
4 x - 4 x ^ { 2 } & 0 \leqslant x \leqslant 0.5
1 & \text { otherwise }
\end{array} \right.$$
Using this model, find \(\mathrm { P } ( X > 0.4 )\)
A random sample of 100 poles cut by this new machine is taken.
Using a suitable approximation, find the probability that at least 8 of these poles have an error of more than 0.4 metres.