| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Single period normal approximation - large lambda direct |
| Difficulty | Standard +0.3 This is a straightforward application of the Poisson distribution with standard calculations (P(X=0) and P(X>7)) plus a normal approximation for large λ. Part (c) requires basic interpretation, and part (d) is a routine normal approximation with continuity correction. All techniques are standard S2 material with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04d Normal approximation to binomial5.02i 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, \(\text{Po}(3.5)\) | B1 |
| (b) (i) \(P(X = 0) = 0.0302\) (from tables) | B1 |
| (ii) \(P(X > 7) = 1 - P(X \leq 7) = 1 - 0.9733 = 0.0267\) | M1 A1 |
| (c) Might not be random – possibly aimed at specific targets | B1 |
| (d) \(\text{Po}(45) \approx N(45, 45)\) | M1 A1 |
| \(P(X > 60) = P(X > 60.5) = P(Z > \frac{15.5/\sqrt{71}}{}) = P(Z > 2.31)\) | M1 A1 A1 |
| \(= 1 - 0.9896 = 0.0104\) | M1 A1 |
| A continuity correction must be made to convert from discrete Poisson to continuous Normal distribution | B2 |
(a) Poisson, $\text{Po}(3.5)$ | B1 |
(b) (i) $P(X = 0) = 0.0302$ (from tables) | B1 |
(ii) $P(X > 7) = 1 - P(X \leq 7) = 1 - 0.9733 = 0.0267$ | M1 A1 |
(c) Might not be random – possibly aimed at specific targets | B1 |
(d) $\text{Po}(45) \approx N(45, 45)$ | M1 A1 |
$P(X > 60) = P(X > 60.5) = P(Z > \frac{15.5/\sqrt{71}}{}) = P(Z > 2.31)$ | M1 A1 A1 |
$= 1 - 0.9896 = 0.0104$ | M1 A1 |
A continuity correction must be made to convert from discrete Poisson to continuous Normal distribution | B2 |
**Total: 14 marks**
---\begin{enumerate}
\item In World War II, the number of V2 missiles that landed on each square mile of London was, on average, $3 \cdot 5$. Assuming that the hits were randomly distributed throughout London,\\
(a) suggest a suitable model for the number of hits on each square mile, giving a suitable value for any parameters.\\
(b) calculate the probability that a particular square mile received\\
(i) no hits,\\
(ii) more than 7 hits.\\
(c) State, with a reason, whether the model is likely to be accurate.
\end{enumerate}
In contrast, the number of bombs weighing more than 1 ton landing on each square mile was 45 .\\
(d) Use a suitable approximation to find the probability that a randomly selected square mile received more than 60 such bombs. Explain what adjustment must be made when using this approximation.\\
\hfill \mbox{\textit{Edexcel S2 Q5 [14]}}