| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2009 |
| Session | January |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Topic | Sum of Poisson processes |
| Type | Minimum time or stock level |
| Difficulty | Moderate -0.3 This is a straightforward application of standard Poisson distribution properties with calculator/tables. Part (a) involves direct probability calculations, (b)(i) tests knowledge that sum of independent Poissons is Poisson, and remaining parts require routine cumulative probability calculations. All techniques are standard S2 material with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.02n Sum of Poisson variables: is Poisson |
3 Joe owns two garages, Acefit and Bestjob, each specialising in the fitting of the latest satellite navigation device.
The daily demand, $X$, for the device at Acefit garage may be modelled by a Poisson distribution with mean 3.6.
The daily demand, $Y$, for the device at Bestjob garage may be modelled by a Poisson distribution with mean 4.4.
\begin{enumerate}[label=(\alph*)]
\item Calculate:
\begin{enumerate}[label=(\roman*)]
\item $\mathrm { P } ( X \leqslant 3 )$;
\item $\quad \mathrm { P } ( Y = 5 )$.
\end{enumerate}\item The total daily demand for the device at Joe's two garages is denoted by $T$.
\begin{enumerate}[label=(\roman*)]
\item Write down the distribution of $T$, stating any assumption that you make.
\item Determine $\mathrm { P } ( 6 < T < 12 )$.
\item Calculate the probability that the total demand for the device will exceed 14 on each of two consecutive days. Give your answer to one significant figure.
\item Determine the minimum number of devices that Joe should have in stock if he is to meet his total demand on at least $99 \%$ of days.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA S2 2009 Q3 [14]}}