Moderate -0.8 This is a straightforward application of standard exponential distribution properties: part (a) requires recalling that mean = 1/λ, and part (b) uses the standard tail probability formula P(X > a) = e^(-λa). Both are direct recall with minimal calculation, making this easier than average.
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.
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.\\
\hfill \mbox{\textit{SPS SPS FM Statistics 2020 Q3 [3]}}