| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2005 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sum of Poisson processes |
| Type | Multi-period repeated application |
| Difficulty | Standard +0.3 This is a straightforward Poisson distribution question requiring basic probability calculations and understanding that Poisson parameters scale with time. Part (a) is direct formula application, part (b)(i) requires knowing that λ scales linearly (2.6×5=13), and part (b)(ii) involves calculating P(Y≥15) then raising to the fourth power for independence. While multi-step, all techniques are standard S2 material with no novel insight required, making it slightly easier than average. |
| Spec | 5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.02n Sum of Poisson variables: is Poisson |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X=2) = \frac{e^{-2.6}(2.6)^2}{2!} = 0.251\) | M1, A1 (2 marks) | \(0.5184 - 0.2674 = 0.251\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(Y \sim P_o(13)\) | B1 (1 mark) | Must state Poisson and 13 |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(Y \geq 15) = 1 - P(Y < 14) = 1 - 0.6751 = 0.3249 \approx 0.325\) | M1, A1\(\checkmark\) (follow through on their \(\lambda\)) | |
| \(\therefore p = (0.3249)^4\) | M\(\hat{1}\) | On their \(p(Y \geq 15)\) |
| \(p = 0.0111\) to \(0.0112\) | (4 marks total) |
## Question 1:
### Part (a)
$P(X=2) = \frac{e^{-2.6}(2.6)^2}{2!} = 0.251$ | M1, A1 (2 marks) | $0.5184 - 0.2674 = 0.251$
### Part (b)(i)
$Y \sim P_o(13)$ | B1 (1 mark) | Must state Poisson **and** 13
### Part (b)(ii)
$P(Y \geq 15) = 1 - P(Y < 14) = 1 - 0.6751 = 0.3249 \approx 0.325$ | M1, A1$\checkmark$ (follow through on their $\lambda$)
$\therefore p = (0.3249)^4$ | M$\hat{1}$ | On their $p(Y \geq 15)$
$p = 0.0111$ to $0.0112$ | (4 marks total)
---1 The number of cars, $X$, passing along a road each minute can be modelled by a Poisson distribution with a mean of 2.6.
\begin{enumerate}[label=(\alph*)]
\item Calculate $\mathrm { P } ( X = 2 )$.
\item \begin{enumerate}[label=(\roman*)]
\item Write down the distribution of $Y$, the number of cars passing along this road in a 5-minute interval.
\item Hence calculate the probability that at least 15 cars pass along this road in each of four successive 5 -minute intervals.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA S2 2005 Q1 [7]}}