| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2006 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sum of Poisson processes |
| Type | Basic sum of two Poissons |
| Difficulty | Standard +0.3 This is a straightforward application of Poisson distribution properties requiring students to scale the parameter for a 5-year period and add independent Poisson variables. The calculations are routine (using tables or calculator) with no conceptual surprises, making it slightly easier than average but still requiring proper understanding of Poisson additivity. |
| Spec | 5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.02n Sum of Poisson variables: is Poisson |
| Answer | Marks | Guidance |
|---|---|---|
| For a 1-year period: Number of A grades \(= \text{Po}(3)\) | B1 | |
| For a 5-year period: Number of A grades \(= \text{Po}(15)\) | B1 | |
| \(P(\text{Total A-grades} > 18) = 1 - P(\text{Total} \leq 18) = 1 - 0.8195 = 0.1805 = 0.181\) | M1, A1 | AWFW 0.180 to 0.181 |
| Answer | Marks | Guidance |
|---|---|---|
| \(X + Y \sim \text{Po}(10)\) | B1 | |
| \(P(X + Y \leq 14) = 0.917\) | M1, A1 | AWFW 0.916 to 0.917 incl |
| Answer | Marks |
|---|---|
| \(X\) and \(Y\) are independent variables | E1 |
## 1(a)
For a 1-year period: Number of A grades $= \text{Po}(3)$ | B1 |
For a 5-year period: Number of A grades $= \text{Po}(15)$ | B1 |
$P(\text{Total A-grades} > 18) = 1 - P(\text{Total} \leq 18) = 1 - 0.8195 = 0.1805 = 0.181$ | M1, A1 | AWFW 0.180 to 0.181
## 1(b)(i)
$X + Y \sim \text{Po}(10)$ | B1 |
$P(X + Y \leq 14) = 0.917$ | M1, A1 | AWFW 0.916 to 0.917 incl
## 1(b)(ii)
$X$ and $Y$ are independent variables | E1 |
---1 The number of A-grades, $X$, achieved in total by students at Lowkey School in their Mathematics examinations each year can be modelled by a Poisson distribution with a mean of 3 .
\begin{enumerate}[label=(\alph*)]
\item Determine the probability that, during a 5 -year period, students at Lowkey School achieve a total of more than 18 A -grades in their Mathematics examinations. (3 marks)
\item The number of A-grades, $Y$, achieved in total by students at Lowkey School in their English examinations each year can be modelled by a Poisson distribution with a mean of 7 .
\begin{enumerate}[label=(\roman*)]
\item Determine the probability that, during a year, students at Lowkey School achieve a total of fewer than 15 A -grades in their Mathematics and English examinations.
\item What assumption did you make in answering part (b)(i)?
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA S2 2006 Q1 [7]}}