| Exam Board | OCR |
|---|---|
| Module | Further Statistics (Further Statistics) |
| Year | 2019 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Three or more independent Poisson sums |
| Difficulty | Standard +0.3 Part (a) requires summing Poisson distributions and using normal approximation—standard Further Maths technique. Part (b) tests conceptual understanding of Poisson conditions (randomness/independence), which is straightforward recall. The calculation is routine for this level, and the explanation requires only basic knowledge of Poisson assumptions. |
| 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 |
|---|---|---|
| 2 | (a) | Po(497) |
| Answer | Marks |
|---|---|
| = 0.1564… | B1 |
| Answer | Marks |
|---|---|
| [3] | 1.1 |
| Answer | Marks |
|---|---|
| 1.1 | Stated or implied |
| Answer | Marks |
|---|---|
| In range [0.156,0.157] | SC: Normal approx.: |
| Answer | Marks | Guidance |
|---|---|---|
| (b) | Occurrence of a bus is not a random event if it | |
| runs on or close to a schedule. | B1 | |
| [1] | 2.4 | Needs context (not just “events”). |
| Answer | Marks |
|---|---|
| between buses is regulated” | Not “not independent” |
Question 2:
2 | (a) | Po(497)
P( 520) = 1 – P( 519) used correctly
= 0.1564… | B1
M1
A1
[3] | 1.1
1.1a
1.1 | Stated or implied
Allow 0.146(08) from 1 – P( 520)
In range [0.156,0.157] | SC: Normal approx.:
N(497, 497) B1
In range [0.156, 0.157]: B2
(b) | Occurrence of a bus is not a random event if it
runs on or close to a schedule. | B1
[1] | 2.4 | Needs context (not just “events”).
Allow just “buses not random”, or
“buses not independent because time
between buses is regulated” | Not “not independent”
without such justification.
Not “not constant rate”.
No extras.2 The average numbers of cars, lorries and buses passing a point on a busy road in a period of 30 minutes are 400, 80 and 17 respectively.
\begin{enumerate}[label=(\alph*)]
\item Assuming that the numbers of each type of vehicle passing the point in a period of 30 minutes have independent Poisson distributions, calculate the probability that the total number of vehicles passing the point in a randomly chosen period of 30 minutes is at least 520.
\item Buses are known to run in approximate accordance with a fixed timetable.
Explain why this casts doubt on the use of a Poisson distribution to model the number of buses passing the point in a fixed time interval.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Statistics 2019 Q2 [4]}}