| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2006 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sum of Poisson processes |
| Type | Multi-period repeated application |
| Difficulty | Moderate -0.3 This is a straightforward application of standard Poisson distribution properties with routine calculations. Part (a) involves basic probability calculation and independence, part (b) tests understanding of scaling the Poisson parameter (λ=9 for 6 months) and cumulative probability, and part (c) requires recall of Poisson assumptions. All parts are textbook-standard 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 probabilities5.02l Poisson conditions: for modelling5.02n Sum of Poisson variables: is Poisson |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X=2) = \frac{e^{-1.5} \times (1.5)^2}{2!} = 0.251\) | M1A1 | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(p = (0.251)^3 = 0.0158\) | M1A1√ | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(Y - P_o(9.0)\) | B1 | 1 mark |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(Y \geq 12) = 1 - P(Y \leq 11) = 1 - 0.8030 = 0.197\) | M1, A1 | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| attacks patients: randomly (p constant) independently | B1, B1 | 2 marks |
**1(a)(i)**
$P(X=2) = \frac{e^{-1.5} \times (1.5)^2}{2!} = 0.251$ | M1A1 | 2 marks
**1(a)(ii)**
$p = (0.251)^3 = 0.0158$ | M1A1√ | 2 marks | on their p from (i)
**1(b)(i)**
$Y - P_o(9.0)$ | B1 | 1 mark
**1(b)(ii)**
$P(Y \geq 12) = 1 - P(Y \leq 11) = 1 - 0.8030 = 0.197$ | M1, A1 | 2 marks
**1(c)**
attacks patients: randomly (p constant) independently | B1, B1 | 2 marks | mean of 1.5 ⇒ p small (B1) (unless very few patients)
**Question 1 Total: 9 marks**
---1 A study undertaken by Goodhealth Hospital found that the number of patients each month, $X$, contracting a particular superbug can be modelled by a Poisson distribution with a mean of 1.5 .
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Calculate $\mathrm { P } ( X = 2 )$.
\item Hence determine the probability that exactly 2 patients will contract this superbug in each of three consecutive months.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Write down the distribution of $Y$, the number of patients contracting this superbug in a given 6-month period.
\item Find the probability that at least 12 patients will contract this superbug during a given 6-month period.
\end{enumerate}\item State two assumptions implied by the use of a Poisson model for the number of patients contracting this superbug.
\end{enumerate}
\hfill \mbox{\textit{AQA S2 2006 Q1 [9]}}