| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2017 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating the Binomial to the Poisson distribution |
| Type | State Poisson approximation with justification |
| Difficulty | Moderate -0.8 This question tests standard bookwork on Poisson approximation conditions (large n, small p) and the additive property of Poisson distributions. Part (a)(i) requires straightforward calculation using tables, (a)(ii) is recall of standard conditions, and (b) applies the direct result that sum of independent Poissons has mean and variance equal to sum of parameters. All parts are routine application of memorized results with no problem-solving or novel insight required. |
| Spec | 2.04d Normal approximation to binomial5.02n Sum of Poisson variables: is Poisson5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(Po(2.54)\) | M1 | seen or implied \(Po(2540 \times 0.001)\) |
| \(1 - e^{-2.54}(1 + 2.54)\) | M1 | any \(\lambda\), allow 1 end error |
| \(= 0.721\) (3 sf) | A1 | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(n\) large and \(p\) small (or \(np\) \((= 2.54) < 5\)) | B1 | \(n > 50\), \(p < 0.1\) |
| Total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\mu = 5.6\) | B1 | |
| \(\sigma = 2.37\) (3 sf) | B1 | Accept \(\sqrt{5.6}\) |
| Total: 2 |
## Question 1:
**Part 1(a)(i):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $Po(2.54)$ | M1 | seen or implied $Po(2540 \times 0.001)$ |
| $1 - e^{-2.54}(1 + 2.54)$ | M1 | any $\lambda$, allow 1 end error |
| $= 0.721$ (3 sf) | A1 | |
| | **Total: 3** | |
**Part 1(a)(ii):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $n$ large and $p$ small (or $np$ $(= 2.54) < 5$) | B1 | $n > 50$, $p < 0.1$ |
| | **Total: 1** | |
**Part 1(b):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\mu = 5.6$ | B1 | |
| $\sigma = 2.37$ (3 sf) | B1 | Accept $\sqrt{5.6}$ |
| | **Total: 2** | |
---1
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item A random variable $X$ has the distribution $\mathrm { B } ( 2540,0.001 )$. Use the Poisson approximation to the binomial distribution to find $\mathrm { P } ( X > 1 )$.
\item Explain why the Poisson approximation is appropriate in this case.
\end{enumerate}\item Two independent random variables, $S$ and $T$, have distributions $\operatorname { Po } ( 2.1 )$ and $\operatorname { Po } ( 3.5 )$ respectively. Find the mean and standard deviation of $S + T$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2017 Q1 [6]}}