| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2023 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating the Binomial to the Poisson distribution |
| Type | Explain E(X) and Var(X) relationship |
| Difficulty | Moderate -0.8 This question tests standard bookwork about Poisson approximation to binomial. Part (a) is pure recall, part (b) requires explaining why np ā np(1-p) needs pā0 (straightforward algebra), and part (c) is a routine calculation with given parameters. All parts are textbook exercises with no problem-solving or novel insight required. |
| Spec | 2.04d Normal approximation to binomial5.02i Poisson distribution: random events model5.02m Poisson: mean = variance = lambda |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(E(W) = \text{Var}(W)\) | B1 | Allow 'they are the same' OE. Must be \(=\) not \(\approx\) (and not both \(=\) and \(\approx\)). Condone \(E(W) = \lambda\) and \(\text{Var}(W) = \lambda\). |
| Total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(np \approx np(1-p)\), hence \(1-p\) must be close to \(1\) | B1 | OE. Must see formulae and \(q = 1-p\) must be seen or implied and conclusion made. |
| Total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\lambda = 1.4\) | B1 | Seen |
| \(1 - e^{-1.4}(1 + 1.4 + \frac{1.4^2}{2})\) or \(1 - e^{-1.4}(1 + 1.4 + 0.98)\) or \(1-(0.2466+0.3452+0.2417)\) | M1 | Allow any \(\lambda\); allow one end error. Expression must be seen (accept correct sigma notation) |
| \(= 0.167\) (3 sf) or \(0.166\) | A1 | Use of Binomial scores SCB1 for 0.167 or 0.166. No working: 0.167 [or 0.166] SC B1. Note: \(\lambda=1.4\) and 0.167 with no working seen scores SC B1B1. Use of Normal scores B0M0 |
**Question 2(a):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $E(W) = \text{Var}(W)$ | B1 | Allow 'they are the same' OE. Must be $=$ not $\approx$ (and not both $=$ and $\approx$). Condone $E(W) = \lambda$ and $\text{Var}(W) = \lambda$. |
| **Total: 1** | | |
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**Question 2(b):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $np \approx np(1-p)$, hence $1-p$ must be close to $1$ | B1 | OE. Must see formulae and $q = 1-p$ must be seen or implied and conclusion made. |
| **Total: 1** | | |
## Question 2(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\lambda = 1.4$ | B1 | Seen |
| $1 - e^{-1.4}(1 + 1.4 + \frac{1.4^2}{2})$ or $1 - e^{-1.4}(1 + 1.4 + 0.98)$ or $1-(0.2466+0.3452+0.2417)$ | M1 | Allow any $\lambda$; allow one end error. Expression must be seen (accept correct sigma notation) |
| $= 0.167$ (3 sf) or $0.166$ | A1 | Use of Binomial scores SCB1 for 0.167 or 0.166. No working: 0.167 [or 0.166] SC B1. Note: $\lambda=1.4$ and 0.167 with no working seen scores SC B1B1. Use of Normal scores B0M0 |
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\begin{enumerate}[label=(\alph*)]
\item The random variable $W$ has a Poisson distribution.\\
State the relationship between $\mathrm { E } ( W )$ and $\operatorname { Var } ( W )$.
\item The random variable $X$ has the distribution $\mathrm { B } ( n , p )$. Jyothi wishes to use a Poisson distribution as an approximate distribution for $X$.
Use the formulae for $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$ to explain why it is necessary for $p$ to be close to 0 for this to be a reasonable approximation.
\item Given that $Y$ has the distribution $\mathrm { B } ( 20000,0.00007 )$, use a Poisson distribution to calculate an estimate of $\mathrm { P } ( Y > 2 )$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2023 Q2 [5]}}