| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2006 |
| Session | January |
| Marks | 8 |
| 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 the Poisson distribution with clearly stated rate parameter. Parts (b) and (d) require direct calculation from the Poisson formula, (c) involves independence of consecutive periods (a standard concept), and (a) is simple model identification. All parts are routine textbook exercises requiring no problem-solving insight, though the multi-part structure and consecutive weeks calculation elevate it slightly above pure recall. |
| Spec | 5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| \(X \sim \text{Po}(1.5)\) | B1 | Need Poisson and \(\lambda\) must be in part (a) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X=2) = \frac{e^{-1.5}1.5^2}{2}\) | M1 | \(\frac{e^\mu \mu^2}{2}\) or \(P(X \leq 2) - P(X \leq 1)\) |
| \(= 0.2510\) | A1 | awrt 0.251 |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X \geq 1) = 1 - P(X=0) = 1 - e^{-1.5} = 0.7769\) | B1 | correct exp awrt 0.777 |
| \(P(\text{at least 1 accident per week for 3 weeks}) = 0.7769^3\) | M1 | \((p)^3\) |
| \(= 0.4689\) | A1 | awrt 0.469; note 0.7769 may be implied |
| Answer | Marks | Guidance |
|---|---|---|
| \(X \sim \text{Po}(3)\) | B1 | may be implied |
| \(P(X>4) = 1 - P(X \leq 4)\) | M1 | |
| \(= 0.1847\) | A1 | awrt 0.1847 |
# Question 2:
## Part (a)
| $X \sim \text{Po}(1.5)$ | B1 | Need Poisson and $\lambda$ must be in part (a) |
## Part (b)
| $P(X=2) = \frac{e^{-1.5}1.5^2}{2}$ | M1 | $\frac{e^\mu \mu^2}{2}$ or $P(X \leq 2) - P(X \leq 1)$ |
| $= 0.2510$ | A1 | awrt 0.251 |
## Part (c)
| $P(X \geq 1) = 1 - P(X=0) = 1 - e^{-1.5} = 0.7769$ | B1 | correct exp awrt 0.777 |
| $P(\text{at least 1 accident per week for 3 weeks}) = 0.7769^3$ | M1 | $(p)^3$ |
| $= 0.4689$ | A1 | awrt 0.469; note 0.7769 may be implied |
## Part (d)
| $X \sim \text{Po}(3)$ | B1 | may be implied |
| $P(X>4) = 1 - P(X \leq 4)$ | M1 | |
| $= 0.1847$ | A1 | awrt 0.1847 |
---2. Accidents on a particular stretch of motorway occur at an average rate of 1.5 per week.
\begin{enumerate}[label=(\alph*)]
\item Write down a suitable model to represent the number of accidents per week on this stretch of motorway.
Find the probability that
\item there will be 2 accidents in the same week,
\item there is at least one accident per week for 3 consecutive weeks,
\item there are more than 4 accidents in a 2 week period.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2006 Q2 [8]}}