| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2012 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Conditional probability with Poisson |
| Difficulty | Standard +0.3 This is a straightforward application of Poisson distribution properties and the Central Limit Theorem. Part (i) involves standard Poisson probability calculations and conditional probability. Part (ii) requires knowing that sample means are approximately normal with mean λ and variance λ/n, then a routine normal probability calculation. All techniques are standard S2 content with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.05a Sample mean distribution: central limit theorem5.05b Unbiased estimates: of population mean and variance |
**(i) (a)**
**M1** $P(X > 3) = 1 - e^{-3.2}\left(1 + 3.2 + \frac{3.2^2}{2!}\right)$
**A1** $= 0.620$ (3 sf)
Allow one end error
**[2]**
**(i) (b)**
**M1** $P(X = 3) = e^{-3.2}\frac{3.2^3}{3!} = 0.22262$
**M1** $P(X=3 \cap X \geq 3) = \frac{P(X=3)}{P(X \geq 3)} = \frac{\text{'0.22262'}}{\text{'0.62010'}}$
**A1** $= 0.359$ (3 sf)
May be implied
**[3]** Their $P(X=3)$; Their $P(X \geq 3)$
**(ii) (a)**
**B1** (Approx) normal with mean $3.2$
**B1** variance $= \frac{3.2}{120}$ or $\frac{2}{75}$ or $0.0267$ (3 sfs) oe
**[2]**
**(ii) (b)**
**M1** $\frac{3.3-3.2}{\sqrt{3.2/120}} = 0.612$
**M1** $\Phi(\text{"0.612"})$
**A1** $= 0.730$ (3 sfs)
Allow with cc attempted
Accept $0.73$
**[3]** Or sd $= \frac{\sqrt{3.2}}{120}$ or $0.163$ (3 sfs) oe
---5 A random variable $X$ has the distribution $\operatorname { Po } ( 3.2 )$.\\
(i) A random value of $X$ is found.
\begin{enumerate}[label=(\alph*)]
\item Find $\mathrm { P } ( X \geqslant 3 )$.
\item Find the probability that $X = 3$ given that $X \geqslant 3$.\\
(ii) Random samples of 120 values of $X$ are taken.\\
(a) Describe fully the distribution of the sample mean.\\
(b) Find the probability that the mean of a random sample of size 120 is less than 3.3.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2012 Q5 [10]}}