| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2009 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Single period normal approximation - scaled period (normal approximation only) |
| Difficulty | Standard +0.3 Part (i) is a straightforward Poisson probability calculation with λ=6.35. Part (ii) requires recognizing when to apply normal approximation to Poisson (large λ=889), then performing a standard normal calculation with continuity correction. Both parts are routine applications of well-practiced techniques with no conceptual surprises, making this slightly easier than average. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation5.02i Poisson distribution: random events model5.02k Calculate Poisson probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\lambda = 6.35\) \(P(> 1) = 1 - e^{-6.35}(1 + 6.35) = 0.987\) | M1, A1 [2] | Attempt at Poisson, \(1 - (P(0) + P(1))\) with their \(\lambda\). Accept \(1 - P(0)\). Correct answer. |
| (ii) \(\lambda = 889\) \(P(> 850) = 1 - \Phi\left(\frac{850.5 - 889}{\sqrt{889}}\right) = \Phi(1.29) \text{ or } \Phi(1.31) \text{ with no cc} = 0.902\) | B1, M1*, M1*dep, A1 [4] | Correct mean. Standardising, with or without continuity correction using their N(889, 889). Condone sd/var mixes. Correct area (> 0.5) with or without cc. Correct answer |
(i) $\lambda = 6.35$ $P(> 1) = 1 - e^{-6.35}(1 + 6.35) = 0.987$ | M1, A1 [2] | Attempt at Poisson, $1 - (P(0) + P(1))$ with their $\lambda$. Accept $1 - P(0)$. Correct answer.
(ii) $\lambda = 889$ $P(> 850) = 1 - \Phi\left(\frac{850.5 - 889}{\sqrt{889}}\right) = \Phi(1.29) \text{ or } \Phi(1.31) \text{ with no cc} = 0.902$ | B1, M1*, M1*dep, A1 [4] | Correct mean. Standardising, with or without continuity correction using their N(889, 889). Condone sd/var mixes. Correct area (> 0.5) with or without cc. Correct answer2 A computer user finds that unwanted emails arrive randomly at a uniform average rate of 1.27 per hour.\\
(i) Find the probability that more than 1 unwanted email arrives in a period of 5 hours.\\
(ii) Find the probability that more than 850 unwanted emails arrive in a period of 700 hours.
\hfill \mbox{\textit{CAIE S2 2009 Q2 [6]}}