Find parameter from given information

7 The waiting time, \(T\) minutes, before a customer is served in a restaurant has distribution function F given by $$\mathrm { F } ( t ) = \begin{cases} 1 - \mathrm { e } ^ { - \lambda t } & t \geqslant 0
0 & t < 0 \end{cases}$$ where \(\lambda\) is a positive constant. The standard deviation of \(T\) is 8 . Find

1. the value of \(\lambda\),
2. the probability that a customer has to wait between 5 and 10 minutes before being served,
3. the median value of \(T\).

3. 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 $$f ( x ) = \left\{ \begin{array} { c c } \lambda e ^ { - \lambda x } & x \geq 0
0 & x < 0 \end{array} \right.$$ The mean waiting time is found to be 5 minutes.
a. State the value of \(\lambda\).
b. Use the model to calculate the probability that a customer has to wait longer that 20 minutes for a response.

2. 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 $$f ( x ) = \left\{ \begin{array} { c c } \lambda e ^ { - \lambda x } & x \geq 0
0 & x < 0 \end{array} \right.$$ The mean waiting time is found to be 5 minutes.
a. State the value of \(\lambda\).
b. Use the model to calculate the probability that a customer has to wait longer that 20 minutes for a response.

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) $$\mathrm { f } ( x ) = \begin{cases} \lambda \mathrm { e } ^ { - \lambda x } & x \geqslant 0
0 & x < 0 \end{cases}$$

1. 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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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:

1. the PDF of the model using the exponential distribution,
2. a possible PDF for the improved model.