SPS SPS FM Statistics 2026 January — Question 6 9 marks

Exam BoardSPS
ModuleSPS FM Statistics (SPS FM Statistics)
Year2026
SessionJanuary
Marks9
TopicExponential Distribution
TypeFind parameter from given information
DifficultyModerate -0.8 This question tests standard exponential distribution properties with straightforward calculations. Part (i) uses the direct relationship between mean and λ (mean = 1/λ), part (ii) applies the standard exponential probability formula P(X > a) = e^(-λa), and part (iii) requires a basic sketch showing truncation at x=15. All components are routine applications of textbook formulas with no problem-solving insight required.
Spec5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf
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.
    1. the PDF of the model using the exponential distribution,
    2. a possible PDF for the improved model.
      [0pt]
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)

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

(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) Sketch the following on the same axis.
\begin{enumerate}[label=(\alph*)]
\item the PDF of the model using the exponential distribution,
\item a possible PDF for the improved model.\\[0pt]

\end{enumerate}

\hfill \mbox{\textit{SPS SPS FM Statistics 2026 Q6 [9]}}
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