| Exam Board | OCR |
|---|---|
| Module | Further Statistics AS (Further Statistics AS) |
| Year | 2019 |
| Session | June |
| Marks | 6 |
| 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 Poisson distribution question requiring scaling the rate parameter (120 per hour → 20 per 10 minutes) and applying standard probability calculations. Part (a) is direct computation, part (b) uses the additive property of independent Poisson variables (sum is Po(40)), and part (c) tests understanding of independence assumptions. All techniques are routine for Further Statistics students with no novel problem-solving required. |
| Spec | 5.02i Poisson distribution: random events model5.02n Sum of Poisson variables: is Poisson |
| Answer | Marks | Guidance |
|---|---|---|
| \(1 - P(\leq 27)\) | M1 | Allow M1 for \(1 - 0.9657 = 0.0343\) |
| \(= 0.0525\) | A1 [2] BC | In range \([0.0524, 0.0525]\) BC |
| Answer | Marks | Guidance |
|---|---|---|
| \(Po(40)\) | M1* | \(Po(2\times \text{their } 20)\) stated or implied |
| \(1 - P(\leq 55)\) | depM1 | Allow M1 for \(1 - P(\leq 56) = 0.00658\) or \(1 - 0.990 = 0.01\) or \(0.0097\) |
| \(= 0.00968\) | A1 [3] | Awrt 0.00968 |
| Answer | Marks | Guidance |
|---|---|---|
| Orders on one day are independent of orders on the other | B1 [1] | Use "orders independent", clearly referred to the two different days, needs context [*not* "events"], and nothing else |
# Question 2:
## Part (a)
$1 - P(\leq 27)$ | **M1** | Allow M1 for $1 - 0.9657 = 0.0343$
$= 0.0525$ | **A1 [2]** BC | In range $[0.0524, 0.0525]$ BC
## Part (b)
$Po(40)$ | **M1*** | $Po(2\times \text{their } 20)$ stated or implied
$1 - P(\leq 55)$ | **depM1** | Allow M1 for $1 - P(\leq 56) = 0.00658$ or $1 - 0.990 = 0.01$ or $0.0097$
$= 0.00968$ | **A1 [3]** | Awrt 0.00968
## Part (c)
Orders on one day are independent of orders on the other | **B1 [1]** | Use "orders independent", clearly referred to the two different days, needs context [*not* "events"], and nothing else | *Not* anything affecting given separate Poissons, such as "orders must be independent" or "constant average rate"
---2 On any day, the number of orders received in one randomly chosen hour by an online supplier can be modelled by the distribution $\operatorname { Po } ( 120 )$.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that at least 28 orders are received in a randomly chosen 10 -minute period.
\item Find the probability that in a randomly chosen 10-minute period on one day and a randomly chosen 10-minute period on the next day a total of at least 56 orders are received.
\item State a necessary assumption for the validity of your calculation in part (b).
\end{enumerate}
\hfill \mbox{\textit{OCR Further Statistics AS 2019 Q2 [6]}}