1 A railway company is investigating operations at a junction where delays often occur. Delays (in minutes) are modelled by the random variable \(T\) with the following cumulative distribution function. $$\mathrm { F } ( t ) = \begin{cases} 0 & t \leqslant 0
1 - \mathrm { e } ^ { - \frac { 1 } { 3 } t } & t > 0 \end{cases}$$

1. Find the median delay and the 90th percentile delay.
2. Derive the probability density function of \(T\). Hence use calculus to find the mean delay.
3. Find the probability that a delay lasts longer than the mean delay. You are given that the variance of \(T\) is 9 .
4. Let \(\bar { T }\) denote the mean of a random sample of 30 delays. Write down an approximation to the distribution of \(\bar { T }\).
5. A random sample of 30 delays is found to have mean 4.2 minutes. Does this cast any doubt on the modelling?