| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 15 |
| 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 Poisson distribution question requiring identification of the distribution, standard probability calculations using tables/calculator, and a basic variance test. All steps are routine S2 techniques with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.02f Measures of average and spread2.02g Calculate mean and standard deviation5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities |
| Answer | Marks |
|---|---|
| (a) Poisson: \(Po(1.5)\) | B1 B1 |
| (b) \(P(X \ge 4) = 1 - P(X \le 3) = 1 - 0.9344 = 0.0656\) | M1 A1 A1 |
| (c) Counts in 50 minutes are \(Po(7.5)\), so \(P(X \le 3) = 0.0591\) | B1 M1 A1 |
| (d) Mean \(= \frac{84}{50} = 1.68\); Var. \(= \frac{226}{50} - 1.68^2 = 1.698\) | B1 M1 A1 |
| Mean \(\approx\) variance; this supports Poisson model, but mean \(> 1.5\) | B1 |
| (e) With mean \(1.68\), \(P(X = 2) = e^{-1.68}\frac{(1.68)^2}{2!} = 0.263\) | M1 M1 A1 |
(a) Poisson: $Po(1.5)$ | B1 B1 |
(b) $P(X \ge 4) = 1 - P(X \le 3) = 1 - 0.9344 = 0.0656$ | M1 A1 A1 |
(c) Counts in 50 minutes are $Po(7.5)$, so $P(X \le 3) = 0.0591$ | B1 M1 A1 |
(d) Mean $= \frac{84}{50} = 1.68$; Var. $= \frac{226}{50} - 1.68^2 = 1.698$ | B1 M1 A1 |
Mean $\approx$ variance; this supports Poisson model, but mean $> 1.5$ | B1 |
(e) With mean $1.68$, $P(X = 2) = e^{-1.68}\frac{(1.68)^2}{2!} = 0.263$ | M1 M1 A1 |
**Total: 15 marks**\begin{enumerate}
\item A sample of radioactive material decays randomly, with an approximate mean of 1.5 counts per minute.\\
(a) Name a distribution that would be suitable for modelling the number of counts per minute.
\end{enumerate}
Give any parameters required for the model.\\
(b) Find the probability of at least 4 counts in a randomly chosen minute.\\
(c) Find the probability of 3 counts or fewer in a random interval lasting 5 minutes.
More careful measurements, over 50 one-minute intervals, give the following data for $x$, the number of counts per minute:
$$\sum x = 84 , \quad \sum x ^ { 2 } = 226$$
(d) Decide whether these data support your answer to part (a).\\
(e) Use the improved data to find probability of exactly two counts in a given one-minute interval.\\
\hfill \mbox{\textit{Edexcel S2 Q6 [15]}}