6.
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)
$$\mathrm { f } ( x ) = \begin{cases} \lambda \mathrm { e } ^ { - \lambda x } & x \geqslant 0
0 & x < 0 \end{cases}$$
- 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) Sketch the following on the same axis.
(a) the PDF of the model using the exponential distribution,
(b) a possible PDF for the improved model.
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