| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Explain or apply conditions in context |
| Difficulty | Moderate -0.8 This question tests recall of standard Poisson conditions and routine application of the distribution. Part (a) requires stating textbook conditions with minimal critical analysis. Parts (b)-(d) involve straightforward calculations: adjusting the rate parameter for different time intervals and applying a normal approximation. No problem-solving insight or novel reasoning is required—purely procedural work below average A-level difficulty. |
| Spec | 5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.02l Poisson conditions: for modelling5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
| Answer | Marks |
|---|---|
| Events must occur singly, at random, at constant rate fairly valid although rate may vary through evening | B2 B1 |
| Let \(X =\) no. of visitors per 10 minutes \(\therefore X \sim Po(5)\) | M1 |
| \(P(X < 2) = P(X \leq 1) = 0.0404\) | M1 A1 |
| Let \(Y =\) no. of visitors per 15 minutes \(\therefore Y \sim Po(7.5)\) | M1 |
| \(P(Y \geq 10) = 1 - P(Y \leq 9) = 1 - 0.7764 = 0.2236\) | M1 A1 |
| Let \(A =\) no. of visitors per 3 hours \(\therefore A \sim Po(90)\) | M1 |
| \(N\) approx. \(B \sim N(90, 90)\) | M1 |
| \(P(A > 100) = P(B > 100.5)\) | M1 |
| \(= P(Z > \frac{100.5 - 90}{\sqrt{90}}) = P(Z > 1.11)\) | A1 |
| \(= 1 - 0.8665 = 0.1335\) | A1 |
Events must occur singly, at random, at constant rate fairly valid although rate may vary through evening | B2 B1 |
Let $X =$ no. of visitors per 10 minutes $\therefore X \sim Po(5)$ | M1 |
$P(X < 2) = P(X \leq 1) = 0.0404$ | M1 A1 |
Let $Y =$ no. of visitors per 15 minutes $\therefore Y \sim Po(7.5)$ | M1 |
$P(Y \geq 10) = 1 - P(Y \leq 9) = 1 - 0.7764 = 0.2236$ | M1 A1 |
Let $A =$ no. of visitors per 3 hours $\therefore A \sim Po(90)$ | M1 |
$N$ approx. $B \sim N(90, 90)$ | M1 |
$P(A > 100) = P(B > 100.5)$ | M1 |
$= P(Z > \frac{100.5 - 90}{\sqrt{90}}) = P(Z > 1.11)$ | A1 |
$= 1 - 0.8665 = 0.1335$ | A1 |
---4. A music website is visited by an average of 30 different people per hour on a weekday evening. The site's designer believes that the number of visitors to the site per hour can be modelled by a Poisson distribution.
\begin{enumerate}[label=(\alph*)]
\item State the conditions necessary for a Poisson distribution to be applicable and comment on their validity in this case.
Assuming that the number of visitors does follow a Poisson distribution, find the probability that there will be
\item less than two visitors in a 10 -minute interval,
\item at least ten visitors in a 15-minute interval.
\item Using a suitable approximation, find the probability of the site being visited by more than 100 people between 6 pm and 9 pm on a Thursday evening.\\
(5 marks)
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q4 [14]}}