4 A survey is carried out into the length of time for which customers wait for a response on a telephone helpline. A statistician who is analysing the results of the survey starts by modelling the waiting time, \(x\) minutes, by an exponential distribution with probability density function (PDF)
$$\mathrm { f } ( x ) = \begin{cases} \lambda \mathrm { e } ^ { - \lambda x } & x \geqslant 0 \\ 0 & x < 0 \end{cases}$$
\section*{(i) In this question you must show detailed reasoning.}
The mean waiting time is found to be 5.0 minutes. Show that \(\lambda = 0.2\).
(ii) Use the model to calculate the probability that a customer has to wait longer than 20 minutes for a response.
In practice it is found that no customer waits for more than 15 minutes for a response. The statistician constructs an improved model to incorporate this fact.
(iii) On the diagram in the Printed Answer Booklet, sketch the following, labelling the curves clearly:
- the PDF of the model using the exponential distribution,
- a possible PDF for the improved model.