| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2015 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sum of Poisson processes |
| Type | Basic sum of two Poissons |
| Difficulty | Standard +0.3 This is a straightforward application of Poisson distribution with standard calculations: (i) and (ii) require basic parameter scaling and probability lookup/calculation, while (iii) involves the standard result that the sum of independent Poisson variables is Poisson. All parts are routine textbook exercises requiring only recall of formulas and calculator work, 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 | Marks | Guidance |
| \(e^{-3.5} \times \frac{3.5^3}{3!}\) | M1 | \(P(X=3)\) any \(\lambda\) |
| \(= 0.216\) (3 sf) | A1 [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(N(42, 42)\) stated or implied | B1 | |
| \(\frac{29.5-42}{\sqrt{42}}\ (= -1.929)\) | M1 | Allow with wrong or no cc OR without \(\sqrt{}\) |
| \(P(z > {}^{\prime}{-1.929}{}^{\prime}) = \Phi({}^{\prime}1.929{}^{\prime}) = 0.973\) (3 sf) | M1, A1 [4] | For correct area consistent with their working |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((\lambda) = 2.4\) | B1 | |
| \(1-e^{-2.4}\left(1+2.4+\frac{2.4^2}{2}+\frac{2.4^3}{3!}\right)\) | M1, M1 | for \(1-P(X \leq 3)\), any \(\lambda\) allow one end error; Correct expression any \(\lambda\) |
| \(= 0.221\) (3 sf) | A1 [4] | NB For combination method: B1 attempting 10 combinations with \(\lambda=1\), \(\lambda=1.4\); M1 6 expressions; M1 10 expressions; 0.221 A1 |
## Question 6:
### Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $e^{-3.5} \times \frac{3.5^3}{3!}$ | M1 | $P(X=3)$ any $\lambda$ |
| $= 0.216$ (3 sf) | A1 [2] | |
### Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $N(42, 42)$ stated or implied | B1 | |
| $\frac{29.5-42}{\sqrt{42}}\ (= -1.929)$ | M1 | Allow with wrong or no cc OR without $\sqrt{}$ |
| $P(z > {}^{\prime}{-1.929}{}^{\prime}) = \Phi({}^{\prime}1.929{}^{\prime}) = 0.973$ (3 sf) | M1, A1 [4] | For correct area consistent with their working |
### Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(\lambda) = 2.4$ | B1 | |
| $1-e^{-2.4}\left(1+2.4+\frac{2.4^2}{2}+\frac{2.4^3}{3!}\right)$ | M1, M1 | for $1-P(X \leq 3)$, any $\lambda$ allow one end error; Correct expression any $\lambda$ |
| $= 0.221$ (3 sf) | A1 [4] | NB For combination method: B1 attempting 10 combinations with $\lambda=1$, $\lambda=1.4$; M1 6 expressions; M1 10 expressions; 0.221 A1 |
---6 People arrive at a checkout in a store at random, and at a constant mean rate of 0.7 per minute. Find the probability that\\
(i) exactly 3 people arrive at the checkout during a 5 -minute period,\\
(ii) at least 30 people arrive at the checkout during a 1-hour period.
People arrive independently at another checkout in the store at random, and at a constant mean rate of 0.5 per minute.\\
(iii) Find the probability that a total of more than 3 people arrive at this pair of checkouts during a 2-minute period.
\hfill \mbox{\textit{CAIE S2 2015 Q6 [10]}}