| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2021 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sum of Poisson processes |
| Type | Basic sum of two Poissons |
| Difficulty | Moderate -0.3 This is a straightforward application of standard Poisson distribution techniques: part (a) requires direct calculation with λ=2, part (b) uses normal approximation (a routine S2 skill), and part (c) applies the sum of independent Poisson variables. All parts follow textbook procedures with no novel problem-solving required, making it slightly easier than average but still requiring competent execution of multiple techniques. |
| Spec | 5.02i Poisson distribution: random events model5.02k Calculate Poisson probabilities5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(e^{-2}\!\left(1 + 2 + \dfrac{2^2}{2!}\right)\) | M1 | \(P(X < 3)\) any \(\lambda\). Allow one end error |
| \(0.677\) (3sf) | A1 | Unsupported correct answer scores SC B1 only |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(N(40, 40)\) | M1 | SOI |
| \(\dfrac{50.5 - 40}{\sqrt{40}}\ [= 1.660]\) | M1 | For standardising with *their* values. Allow with wrong or no cc; must have square root |
| \(P(z > \text{'1.660'}) = 1 - \Phi(\text{'1.660'})\) | M1 | Correct area consistent with *their* working |
| \(0.0485\) or \(0.0484\) (3sf) | A1 | |
| 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\lambda = 10\) | B1 | Condone mean \(= 10\) |
| \(e^{-10}\!\left(\dfrac{10^8}{8!} + \dfrac{10^9}{9!} + \dfrac{10^{10}}{10!} + \dfrac{10^{11}}{11!}\right)\) | M1 | Allow any \(\lambda\) (allow one end error) |
| \(0.477\) (3sf) | A1 | Unsupported correct answer scores SC B2 only |
| 3 |
## Question 5(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $e^{-2}\!\left(1 + 2 + \dfrac{2^2}{2!}\right)$ | **M1** | $P(X < 3)$ any $\lambda$. Allow one end error |
| $0.677$ (3sf) | **A1** | Unsupported correct answer scores SC **B1** only |
| | **2** | |
---
## Question 5(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $N(40, 40)$ | **M1** | SOI |
| $\dfrac{50.5 - 40}{\sqrt{40}}\ [= 1.660]$ | **M1** | For standardising with *their* values. Allow with wrong or no cc; must have square root |
| $P(z > \text{'1.660'}) = 1 - \Phi(\text{'1.660'})$ | **M1** | Correct area consistent with *their* working |
| $0.0485$ or $0.0484$ (3sf) | **A1** | |
| | **4** | |
---
## Question 5(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\lambda = 10$ | **B1** | Condone mean $= 10$ |
| $e^{-10}\!\left(\dfrac{10^8}{8!} + \dfrac{10^9}{9!} + \dfrac{10^{10}}{10!} + \dfrac{10^{11}}{11!}\right)$ | **M1** | Allow any $\lambda$ (allow one end error) |
| $0.477$ (3sf) | **A1** | Unsupported correct answer scores SC **B2** only |
| | **3** | |
---5 In a certain large document, typing errors occur at random and at a constant mean rate of 0.2 per page.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that there are fewer than 3 typing errors in 10 randomly chosen pages.
\item Use an approximating distribution to find the probability that there are more than 50 typing errors in 200 randomly chosen pages.\\
In the same document, formatting errors occur at random and at a constant mean rate of 0.3 per page.
\item Find the probability that the total number of typing and formatting errors in 20 randomly chosen pages is between 8 and 11 inclusive.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2021 Q5 [9]}}