| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2022 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sum of Poisson processes |
| Type | Rescale rate then sum Poissons |
| Difficulty | Standard +0.3 This is a straightforward application of Poisson distribution with standard techniques: part (a) requires rate adjustment and direct calculation, part (b) uses normal approximation (a standard S2 topic), and part (c) combines independent Poisson variables (sum property). All steps are routine textbook exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04d Normal approximation to binomial5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\lambda = 4.5\) | B1 | |
| \(1 - e^{-4.5}\left(1 + 4.5 + \frac{4.5^2}{2!} + \frac{4.5^3}{3!} + \frac{4.5^4}{4!}\right)\) | M1 | Allow one end error; Allow any \(\lambda\); Poisson expressions must be seen |
| \(= 0.468\) (3 sf) | A1 | If M0 awarded allow SC B1 for 0.468 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\lambda = 162\); \(X \sim \text{Po}(162) \Rightarrow X \sim N(162, 162)\) | B1 | |
| \(\frac{149.5 - 162}{\sqrt{162}}\) and \(\frac{160.5 - 162}{\sqrt{162}}\) \((= -0.982\) and \(-0.118)\) | M1 | One of these; allow with incorrect or no continuity correction |
| \(\Phi(0.982) - \Phi(0.118)\) oe | M1 | Area consistent with *their* values (both standardisations must be seen) |
| \(= 0.290\) (3 sf) | A1 | Allow 0.29 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\lambda = \frac{13.5}{6} + 3.6 \times \frac{2}{3}\) OE or \(4.65\) | M1 | Attempt to find \(\lambda\) |
| \(e^{-4.65}\left(\frac{4.65^4}{4!} + \frac{4.65^5}{5!} + \frac{4.65^6}{6!}\right)\) | M1 | Allow any \(\lambda\); Allow one end error; Poisson terms not be seen |
| \(0.494\) (3 sf) | A1 | If M0 allow SC B1 for 0.494 |
## Question 5(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\lambda = 4.5$ | B1 | |
| $1 - e^{-4.5}\left(1 + 4.5 + \frac{4.5^2}{2!} + \frac{4.5^3}{3!} + \frac{4.5^4}{4!}\right)$ | M1 | Allow one end error; Allow any $\lambda$; Poisson expressions must be seen |
| $= 0.468$ (3 sf) | A1 | If M0 awarded allow SC B1 for 0.468 |
---
## Question 5(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\lambda = 162$; $X \sim \text{Po}(162) \Rightarrow X \sim N(162, 162)$ | B1 | |
| $\frac{149.5 - 162}{\sqrt{162}}$ and $\frac{160.5 - 162}{\sqrt{162}}$ $(= -0.982$ and $-0.118)$ | M1 | One of these; allow with incorrect or no continuity correction |
| $\Phi(0.982) - \Phi(0.118)$ oe | M1 | Area consistent with *their* values (both standardisations must be seen) |
| $= 0.290$ (3 sf) | A1 | Allow 0.29 |
---
## Question 5(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\lambda = \frac{13.5}{6} + 3.6 \times \frac{2}{3}$ OE or $4.65$ | M1 | Attempt to find $\lambda$ |
| $e^{-4.65}\left(\frac{4.65^4}{4!} + \frac{4.65^5}{5!} + \frac{4.65^6}{6!}\right)$ | M1 | Allow any $\lambda$; Allow one end error; Poisson terms not be seen |
| $0.494$ (3 sf) | A1 | If M0 allow SC B1 for 0.494 |
---5 Cars arrive at a fuel station at random and at a constant average rate of 13.5 per hour.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that more than 4 cars arrive during a 20-minute period.
\item Use an approximating distribution to find the probability that the number of cars that arrive during a 12-hour period is between 150 and 160 inclusive.\\
Independently of cars, trucks arrive at the fuel station at random and at a constant average rate of 3.6 per 15-minute period.
\item Find the probability that the total number of cars and trucks arriving at the fuel station during a 10 -minute period is more than 3 and less than 7 .
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2022 Q5 [10]}}