| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2006 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sum of Poisson processes |
| Type | Normal approximation to summed Poisson |
| Difficulty | Standard +0.3 This question tests standard Poisson distribution properties: scaling the rate parameter for different populations and applying normal approximation for large λ. Part (i) requires recognizing that independent Poisson variables sum to another Poisson, then a straightforward calculation. Part (ii) is a textbook application of Poisson→Normal approximation with continuity correction. Both parts are routine applications of well-practiced techniques with no novel problem-solving required, making this slightly easier than average. |
| Spec | 5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02n Sum of Poisson variables: is Poisson5.04b Linear combinations: of normal distributions |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\lambda_1 = 3.6\) | M1 | Attempt at using Poisson with a different mean |
| \(\lambda_2 = 4.2\) | M1 | An attempt at P(3) using their 7.8 |
| \(P(3) = e^{-7.8} \times \frac{7.8^3}{3!} = 0.0324\) | A1 | Correct answer |
| (ii) \(\lambda = 64 \times 6 = 384\) | M1 | Their variance = their mean (with attempt at 384) |
| \(X \sim N(384, 384)\) | M1 | Standardising, with or without cc |
| \(P(X < 369) = \Phi\left(\frac{368.5 - 384}{\sqrt{384}}\right)\) | B1 | Correct cc within a std expression |
| \(= \Phi(-0.791) = 1 - 0.7855 = 0.215\) | A1 | Correct answer, accept 0.214 (cwo) |
**(i)** $\lambda_1 = 3.6$ | M1 | Attempt at using Poisson with a different mean
$\lambda_2 = 4.2$ | M1 | An attempt at P(3) using their 7.8
$P(3) = e^{-7.8} \times \frac{7.8^3}{3!} = 0.0324$ | A1 | Correct answer | 3 marks total
**(ii)** $\lambda = 64 \times 6 = 384$ | M1 | Their variance = their mean (with attempt at 384)
$X \sim N(384, 384)$ | M1 | Standardising, with or without cc
$P(X < 369) = \Phi\left(\frac{368.5 - 384}{\sqrt{384}}\right)$ | B1 | Correct cc within a std expression
$= \Phi(-0.791) = 1 - 0.7855 = 0.215$ | A1 | Correct answer, accept 0.214 (cwo) | 4 marks total
---4 In summer, wasps' nests occur randomly in the south of England at an average rate of 3 nests for every 500 houses.\\
(i) Find the probability that two villages in the south of England, with 600 houses and 700 houses, have a total of exactly 3 wasps' nests.\\
(ii) Use a suitable approximation to estimate the probability of there being fewer than 369 wasps' nests in a town with 64000 houses.
\hfill \mbox{\textit{CAIE S2 2006 Q4 [7]}}