| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2002 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Single period normal approximation - scaled period (normal approximation only) |
| Difficulty | Standard +0.3 This is a straightforward application of Poisson distribution with standard techniques: part (i) uses the basic Poisson formula, part (ii) requires adjusting the rate parameter for a different time period, and part (iii) involves a routine normal approximation to Poisson. All steps are textbook procedures with no novel problem-solving required, 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 probabilities5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X = 5) = 0.161\) | M1, A1 (2 marks) | For mean 6 and evaluating a Poisson prob; For correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| \(1 - P(0) = 0.865\) | B1, M1, A1 (3 marks) | For \(\mu = 2\) used in a Poisson prob.; For \(1 - P(0)\), any mean; For correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| \(1 - \Phi(0.9186) = 0.179\) | B1, B1, M1, A1, A1 (5 marks) | For \(\mu = 24\); For their var=their mean; For standardising with or without cc; For correct continuity correction; For correct answer (SR Using Poisson with no approximation (0.180(26)) scores M1 A1 only) |
**(i)** $\mu\text{can} = 6$
$P(X = 5) = 0.161$ | M1, A1 (2 marks) | For mean 6 and evaluating a Poisson prob; For correct answer
**(ii)** $\mu = 2$
$P(0) = e^{-2}(= 0.135)$
$1 - P(0) = 0.865$ | B1, M1, A1 (3 marks) | For $\mu = 2$ used in a Poisson prob.; For $1 - P(0)$, any mean; For correct answer
**(iii)** $\mu = 24$, $\sigma^2 = 24$
$z = \frac{19.5 - 24}{\sqrt{24}} = -0.9186$
$1 - \Phi(0.9186) = 0.179$ | B1, B1, M1, A1, A1 (5 marks) | For $\mu = 24$; For their var=their mean; For standardising with or without cc; For correct continuity correction; For correct answer (SR Using Poisson with no approximation (0.180(26)) scores M1 A1 only)
---6 Between 7 p.m. and 11 p.m., arrivals of patients at the casualty department of a hospital occur at random at an average rate of 6 per hour.\\
(i) Find the probability that, during any period of one hour between 7 p.m. and 11 p.m., exactly 5 people will arrive.\\
(ii) A patient arrives at exactly 10.15 p.m. Find the probability that at least one more patient arrives before 10.35 p.m.\\
(iii) Use a suitable approximation to estimate the probability that fewer than 20 patients arrive at the casualty department between 7 p.m. and 11 p.m. on any particular night.
\hfill \mbox{\textit{CAIE S2 2002 Q6 [10]}}