| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2007 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sum of Poisson processes |
| Type | Multi-period repeated application |
| Difficulty | Moderate -0.3 This is a straightforward S2 question testing standard Poisson distribution techniques: direct probability calculations, sum of independent Poisson variables (which students learn forms another Poisson), and binomial probability with the result from part (a). The mean/variance comparison in part (c) is textbook material. All parts are routine applications with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.02i Poisson distribution: random events model5.02j 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 |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(P(A=4) = \dfrac{e^{-3.5} \times (3.5)^4}{4!} = 0.189\) | M1A1 | Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(P(B \leq 6) = 0.762\) | B1 | Total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(T = A + B \sim \text{Po}(8.5)\) | ||
| \(P(T \text{ fewer than } 10) = P(T < 10)\) | M1 | Use of Po\((8.5)\) |
| \(= P(T \leq 9)\) | M1 | \(T \leq 9\) attempted |
| \(= 0.653\) | A1 | CAO, Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(X \sim B(5, 0.653)\) | B1 | \(X \sim B(5, \text{their } p)\) |
| \(P(X \geq 4) = \dbinom{5}{4}(0.653)^4(0.347) + (0.653)^5\) | M1 | |
| \(= 0.31547 + 0.11873 = 0.434\) | A1\(\checkmark\) | On their \(p\) from (a)(iii), Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\bar{x} = 9.2\) | B1 | |
| \(s^2 = 9.29\) | B1 | \(\sigma^2 = 8.36\), Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Mean and variance have similar values which suggests that Poisson distribution may be appropriate | B1\(\checkmark\) B1\(\checkmark\) | Total: 2 |
## Question 2:
### Part (a)(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $P(A=4) = \dfrac{e^{-3.5} \times (3.5)^4}{4!} = 0.189$ | M1A1 | **Total: 2** |
### Part (a)(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $P(B \leq 6) = 0.762$ | B1 | **Total: 1** |
### Part (a)(iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $T = A + B \sim \text{Po}(8.5)$ | | |
| $P(T \text{ fewer than } 10) = P(T < 10)$ | M1 | Use of Po$(8.5)$ |
| $= P(T \leq 9)$ | M1 | $T \leq 9$ attempted |
| $= 0.653$ | A1 | CAO, **Total: 3** |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $X \sim B(5, 0.653)$ | B1 | $X \sim B(5, \text{their } p)$ |
| $P(X \geq 4) = \dbinom{5}{4}(0.653)^4(0.347) + (0.653)^5$ | M1 | |
| $= 0.31547 + 0.11873 = 0.434$ | A1$\checkmark$ | On their $p$ from (a)(iii), **Total: 3** |
### Part (c)(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\bar{x} = 9.2$ | B1 | |
| $s^2 = 9.29$ | B1 | $\sigma^2 = 8.36$, **Total: 2** |
### Part (c)(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Mean and variance have similar values which suggests that Poisson distribution may be appropriate | B1$\checkmark$ B1$\checkmark$ | **Total: 2** |
---2 The number of computers, $A$, bought during one day from the Amplebuy computer store can be modelled by a Poisson distribution with a mean of 3.5.
The number of computers, $B$, bought during one day from the Bestbuy computer store can be modelled by a Poisson distribution with a mean of 5.0 .
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Calculate $\mathrm { P } ( A = 4 )$.
\item Determine $\mathrm { P } ( B \leqslant 6 )$.
\item Find the probability that a total of fewer than 10 computers is bought from these two stores on one particular day.
\end{enumerate}\item Calculate the probability that a total of fewer than 10 computers is bought from these two stores on at least 4 out of 5 consecutive days.
\item The numbers of computers bought from the Choicebuy computer store over a 10-day period are recorded as
$$\begin{array} { l l l l l l l l l l }
8 & 12 & 6 & 6 & 9 & 15 & 10 & 8 & 6 & 12
\end{array}$$
\begin{enumerate}[label=(\roman*)]
\item Calculate the mean and variance of these data.
\item State, giving a reason based on your results in part (c)(i), whether or not a Poisson distribution provides a suitable model for these data.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA S2 2007 Q2 [13]}}