| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2010 |
| Session | November |
| Marks | 5 |
| 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 clear rate conversions. Part (i) requires simple parameter scaling (2/3 per minute × 5 minutes) and direct probability calculation. Part (ii) involves summing two independent Poisson variables and computing P(X<3), which is standard bookwork requiring only basic probability addition. No conceptual challenges beyond routine application of formulas. |
| Spec | 5.02i Poisson distribution: random events model5.02k Calculate Poisson probabilities5.02n Sum of Poisson variables: is Poisson |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(e^{-\frac{10}{3}} \times \frac{(\frac{10}{3})^4}{4!}\) | M1 | Allow incorrect \(\lambda\) |
| \(= 0.184\) or \(0.183\) | A1 [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\lambda = 5\) | B1 | |
| \(e^{-5}(1 + 5 + \frac{5^2}{2})\) | M1 | Allow incorrect \(\lambda\). Allow one end error |
| \(= 0.125\) (3 sfs) | A1 [3] | OR Combination method scores B1, identifying all 6 possible combinations M1, multiply each combination and add (must use at least 5 combinations) A1 |
## Question 2:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $e^{-\frac{10}{3}} \times \frac{(\frac{10}{3})^4}{4!}$ | M1 | Allow incorrect $\lambda$ |
| $= 0.184$ or $0.183$ | A1 [2] | |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\lambda = 5$ | B1 | |
| $e^{-5}(1 + 5 + \frac{5^2}{2})$ | M1 | Allow incorrect $\lambda$. Allow one end error |
| $= 0.125$ (3 sfs) | A1 [3] | OR Combination method scores B1, identifying all 6 possible combinations M1, multiply each combination and add (must use at least 5 combinations) A1 |
---2 People arrive randomly and independently at a supermarket checkout at an average rate of 2 people every 3 minutes.\\
(i) Find the probability that exactly 4 people arrive in a 5 -minute period.
At another checkout in the same supermarket, people arrive randomly and independently at an average rate of 1 person each minute.\\
(ii) Find the probability that a total of fewer than 3 people arrive at the two checkouts in a 3 -minute period.
\hfill \mbox{\textit{CAIE S2 2010 Q2 [5]}}