| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Single period normal approximation - scaled period (normal approximation only) |
| Difficulty | Moderate -0.3 This is a straightforward Poisson distribution question with standard parts: (a) direct probability calculation, (b) monthly rate adjustment, and (c) normal approximation to Poisson. All parts follow textbook procedures with no conceptual challenges—students need only identify the distribution, adjust parameters appropriately, and apply standard formulas or approximations. Slightly easier than average due to clear setup and routine application. |
| Spec | 5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.04b Linear combinations: of normal distributions |
| Answer | Marks | Guidance |
|---|---|---|
| (a) let \(X\) = no. of donations over £10000 per year \(\therefore X \sim \text{Po}(25)\); \(P(X = 30) = \frac{e^{-25} \times 25^{30}}{30!} = 0.0454\) (3sf) | M1, M1, A1 | |
| (b) let \(Y\) = no. of donations over £10000 per month \(\therefore Y \sim \text{Po}(\frac{25}{12})\); \(P(Y < 3) = P(Y \leq 2) = e^{-\frac{25}{12}}(1 + \frac{25}{12} + \frac{(\frac{25}{12})^2}{2}) = 0.6541\) (4sf) | M1, M1, M1, A1, A1 | |
| (c) let \(D\) = no. of donations over £10000 per 2 years \(\therefore D \sim \text{Po}(50)\); \(N \approx E \sim N(50, 50)\); \(P(D > 45) = P(E > 45.5) = P(Z > \frac{45.5 - 50}{\sqrt{50}}) = P(Z > -0.64) = 0.7389\) | M1, M1, M1, A1, A1 | (13 marks) |
**(a)** let $X$ = no. of donations over £10000 per year $\therefore X \sim \text{Po}(25)$; $P(X = 30) = \frac{e^{-25} \times 25^{30}}{30!} = 0.0454$ (3sf) | M1, M1, A1 |
**(b)** let $Y$ = no. of donations over £10000 per month $\therefore Y \sim \text{Po}(\frac{25}{12})$; $P(Y < 3) = P(Y \leq 2) = e^{-\frac{25}{12}}(1 + \frac{25}{12} + \frac{(\frac{25}{12})^2}{2}) = 0.6541$ (4sf) | M1, M1, M1, A1, A1 |
**(c)** let $D$ = no. of donations over £10000 per 2 years $\therefore D \sim \text{Po}(50)$; $N \approx E \sim N(50, 50)$; $P(D > 45) = P(E > 45.5) = P(Z > \frac{45.5 - 50}{\sqrt{50}}) = P(Z > -0.64) = 0.7389$ | M1, M1, M1, A1, A1 | (13 marks)A charity receives donations of more than £10000 at an average rate of 25 per year.
Find the probability that the charity receives
\begin{enumerate}[label=(\alph*)]
\item exactly 30 such donations in one year, [3]
\item less than 3 such donations in one month. [5]
\item Using a suitable approximation, find the probability that the charity receives more than 45 donations of more than £10000 in the next two years. [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q5 [13]}}