| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Poisson with binomial combination |
| Difficulty | Moderate -0.3 This is a straightforward S2 question testing standard Poisson distribution concepts. Parts (a)-(c) involve routine calculations of mean/variance from a frequency table and recognizing Poisson properties (mean ≈ variance). Parts (d)-(e) require basic Poisson probability calculations and applying binomial distribution. All techniques are standard textbook exercises with no novel problem-solving required, making it slightly easier than average A-level difficulty. |
| Spec | 5.02i Poisson distribution: random events model5.02k Calculate Poisson probabilities5.02l Poisson conditions: for modelling |
| \(X\) | 0 | 1 | 2 | 3 | 4 | 5 |
| \(f\) | 3 | 8 | 5 | 3 | 2 | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Mean \(= \frac{40}{22} = 1.82\) Variance \(= \frac{112}{22} - 1.82^2 = 1.79\) | M1 A1 M1 A1 | |
| (b) mean \(\approx\) variance | B1 | |
| (c) positive skewness | B1 | |
| (d) \(P(X < 2) = e^{-2}(1 + 2 \cdot 4) = 0.308\) | M1 A1 A1 | |
| (e) \(^{22}C_{11}(0.308)^{11}(0.692)^{11} = 0.0293\) | M1 A1 A1 | Total: 12 marks |
(a) Mean $= \frac{40}{22} = 1.82$ Variance $= \frac{112}{22} - 1.82^2 = 1.79$ | M1 A1 M1 A1 |
(b) mean $\approx$ variance | B1 |
(c) positive skewness | B1 |
(d) $P(X < 2) = e^{-2}(1 + 2 \cdot 4) = 0.308$ | M1 A1 A1 |
(e) $^{22}C_{11}(0.308)^{11}(0.692)^{11} = 0.0293$ | M1 A1 A1 | **Total: 12 marks**In a survey of 22 families, the number of children, $X$, in each family was given by the following
table, where $f$ denotes the frequency:
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
$X$ & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline
$f$ & 3 & 8 & 5 & 3 & 2 & 1 \\
\hline
\end{tabular}
\begin{enumerate}[label=(\alph*)]
\item Find the mean and variance of $X$. [4 marks]
\item Explain why these results suggest that $X$ may follow a Poisson distribution. [1 mark]
\item State another feature of the data that suggests a Poisson distribution. [1 mark]
\end{enumerate}
It is sometimes suggested that the number of children in a family follows a Poisson distribution
with mean 2·4. Assuming that this is correct,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item find the probability that a family has less than two children. [3 marks]
\item Use this result to find the probability that, in a random sample of 22 families, exactly 11 of
the families have less than two children. [3 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q5 [12]}}