| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Mean-variance comparison for Poisson validation |
| Difficulty | Standard +0.3 This is a straightforward S2 question testing standard Poisson distribution concepts: calculating mean/variance from grouped data, verifying the Poisson property (mean ≈ variance), and using Poisson tables or formulas for probability calculations. All steps are routine applications of learned techniques with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.02i Poisson distribution: random events model5.02k Calculate Poisson probabilities5.02l Poisson conditions: for modelling |
| No of counts, \(X\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| Frequency | 10 | 24 | 29 | 16 | 12 | 6 | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Mean \(= \frac{226}{100} = 2.26\) | M1, A1 | |
| Variance \(= \frac{734}{100} - 2.26^2 = 2.23\) | M1, A1 | |
| Since mean \(\approx\) variance, distribution could be Poisson | B1 | |
| (b) In \(Po(2.26)\), \(P(X > 6) = 0.00623 + 0.00176 + 0.00044 + \ldots = 0.009\) | M1, M1, A1, A1 | |
| (c) In a sample of 100, expected no. \(= 100 \times 0.009 = 0.9\) intervals | M1, A1 | 11 marks total |
(a) Mean $= \frac{226}{100} = 2.26$ | M1, A1 |
Variance $= \frac{734}{100} - 2.26^2 = 2.23$ | M1, A1 |
Since mean $\approx$ variance, distribution could be Poisson | B1 |
(b) In $Po(2.26)$, $P(X > 6) = 0.00623 + 0.00176 + 0.00044 + \ldots = 0.009$ | M1, M1, A1, A1 |
(c) In a sample of 100, expected no. $= 100 \times 0.009 = 0.9$ intervals | M1, A1 | 11 marks total |A Geiger counter is observed in the presence of a radioactive source.
In 100 one-minute intervals, the number of counts recorded are as follows:
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
No of counts, $X$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
Frequency & 10 & 24 & 29 & 16 & 12 & 6 & 3 \\
\hline
\end{tabular}
\begin{enumerate}[label=(\alph*)]
\item Find the mean and variance of this data, and show that it supports the idea that the random variable $X$ is following a Poisson distribution. [5 marks]
\item Use a Poisson distribution with the mean found in part (a) to calculate, to 3 decimal places, the probability that more than 6 counts will be recorded in any particular minute. [4 marks]
\item Find the number of one-minute intervals, in the sample of 100, in which more than 6 counts would be expected. [2 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q4 [11]}}