| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2016 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sum of Poisson processes |
| Type | Joint probability of independent events |
| Difficulty | Standard +0.3 This is a straightforward application of Poisson distribution with two independent processes. Part (i) requires calculating P(X≥2)P(Y≥3) using complement rule, and part (ii) combines the processes (λ=30 for 60 minutes) with a single probability calculation. The question involves standard Poisson techniques with no conceptual challenges beyond recognizing independence and rate scaling, making it slightly easier than average. |
| Spec | 5.02i Poisson distribution: random events model5.02j 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 |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(1 - e^{-1}(1 + 1) \quad (= 0.26424)\) | B1 | B1 for either \(\lambda\) correct |
| \(1 - e^{-1.5}(1 + 1.5 + \frac{1.5^2}{2!}) \quad (= 0.19115)\) | B1 | B1 for either correct expression with correct \(\lambda\) |
| \(\text{'0.26424'} \times \text{'0.19115'}\) | M1 | product of their values for \(\leqslant 2\) and \(\leqslant 3\) from Poisson; need correct form "\(1 - \)…", but allow incorrect \(\lambda\) values and end errors |
| \(= 0.0505\) (3 sf) | A1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\lambda = 30\); \(N(30, 30)\) | B1, B1\(\checkmark\) | seen or implied, need \(N(\lambda, \lambda)\) |
| \(\frac{35.5 - 30}{\sqrt{30}} \quad (= 1.004)\) | M1 | allow with wrong or no cc or no \(\sqrt{\phantom{x}}\) |
| \(\Phi(\text{'1.004'})\) | M1 | consistent with their working |
| \(= 0.842\) (3 sf) | A1 | [5] |
## Question 7:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $1 - e^{-1}(1 + 1) \quad (= 0.26424)$ | B1 | B1 for either $\lambda$ correct |
| $1 - e^{-1.5}(1 + 1.5 + \frac{1.5^2}{2!}) \quad (= 0.19115)$ | B1 | B1 for either correct expression with correct $\lambda$ |
| $\text{'0.26424'} \times \text{'0.19115'}$ | M1 | product of their values for $\leqslant 2$ and $\leqslant 3$ from Poisson; need correct form "$1 - $…", but allow incorrect $\lambda$ values and end errors |
| $= 0.0505$ (3 sf) | A1 | [4] | accept 0.0504 |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\lambda = 30$; $N(30, 30)$ | B1, B1$\checkmark$ | seen or implied, need $N(\lambda, \lambda)$ |
| $\frac{35.5 - 30}{\sqrt{30}} \quad (= 1.004)$ | M1 | allow with wrong or no cc or no $\sqrt{\phantom{x}}$ |
| $\Phi(\text{'1.004'})$ | M1 | consistent with their working |
| $= 0.842$ (3 sf) | A1 | [5] |
---7 Men arrive at a clinic independently and at random, at a constant mean rate of 0.2 per minute. Women arrive at the same clinic independently and at random, at a constant mean rate of 0.3 per minute.\\
(i) Find the probability that at least 2 men and at least 3 women arrive at the clinic during a 5 -minute period.\\
(ii) Find the probability that fewer than 36 people arrive at the clinic during a 1-hour period.
\hfill \mbox{\textit{CAIE S2 2016 Q7 [9]}}