| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2017 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating the Binomial to the Poisson distribution |
| Type | Mean and variance calculations |
| Difficulty | Moderate -0.3 This is a straightforward application of binomial-to-Poisson approximation with standard calculations. Part (i) requires simple multiplication for mean/variance, part (ii) tests understanding of when Poisson approximation is valid (n large, p small, np moderate), and part (iii) is a routine cumulative probability lookup. All steps are textbook procedures with no problem-solving or novel insight required, making it slightly easier than average. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04d Normal approximation to binomial5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(E(X) = 4.197\) | B1 | |
| \(\text{Var}(X) = 4.196\) | B1 | Both to 3dp or better |
| Total: | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(E(X) \approx \text{Var}(X)\) | B1 | Condone \(=\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(e^{-4.1968}\left(1 + 4.1968 + \frac{4.1968^2}{2} + \frac{4.1968^3}{3!} + \frac{4.1968^4}{4!}\right)\) | M1 | Any \(\lambda\). Allow with one end error |
| \(= 0.590\) (3 sf) | A1 | Allow 0.591 |
| Total: | 2 |
## Question 2(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $E(X) = 4.197$ | B1 | |
| $\text{Var}(X) = 4.196$ | B1 | Both to 3dp or better |
| **Total:** | **4** | |
## Question 2(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $E(X) \approx \text{Var}(X)$ | B1 | Condone $=$ |
## Question 2(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $e^{-4.1968}\left(1 + 4.1968 + \frac{4.1968^2}{2} + \frac{4.1968^3}{3!} + \frac{4.1968^4}{4!}\right)$ | M1 | Any $\lambda$. Allow with one end error |
| $= 0.590$ (3 sf) | A1 | Allow 0.591 |
| **Total:** | **2** | |2 Javier writes an article containing 52460 words. He plans to upload the article to his website, but he knows that this process sometimes introduces errors. He assumes that for each word in the uploaded version of his article, the probability that it contains an error is 0.00008 . The number of words containing an error is denoted by $X$.\\
(i) Find $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$, giving your answers correct to three decimal places.\\
Javier wants to use the Poisson distribution as an approximating distribution to calculate the probability that there will be fewer than 5 words containing an error in his uploaded article.\\
(ii) Explain how your answers to part (i) are consistent with the use of the Poisson distribution as an approximating distribution.\\
(iii) Use the Poisson distribution to calculate $\mathrm { P } ( X < 5 )$.\\
\hfill \mbox{\textit{CAIE S2 2017 Q2 [5]}}