| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2013 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Poisson parameter from given probability |
| Difficulty | Standard +0.3 This is a standard S2 Poisson distribution question requiring explanation of conditions (bookwork), contextual discussion (straightforward), and routine algebraic manipulation using P(X=r) formula. The algebra in part (iii) is simple (equating two probabilities leads directly to λ=7.5), making this slightly easier than average A-level. |
| 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 modelling |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Notes |
| There is no pattern to the failures and they occur independently of one another | B0 | |
| Equally likely to occur at any moment in time | B0 | |
| Impossible to predict | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Notes |
| Failures occur singly, unlikely as there could be a power failure that affects all lights in an area | B0B1 | |
| Failures occur independently of each other | (B1) | |
| Likely because one failure does not cause another | B1 | |
| Mean number of traffic light failures is constant each day | B0 | *OK if each hour etc* |
| Failures occur at constant average rate | (B1) | |
| Unlikely as could change with season | B0 | |
| Likely as each set has same probability of failing | B0 | |
| Likely as they run in the same mode all day | B1 |
# Question 8 Specimen Answers
## Question 8(i):
| Answer | Marks | Notes |
|--------|-------|-------|
| There is no pattern to the failures and they occur independently of one another | B0 | |
| Equally likely to occur at any moment in time | B0 | |
| Impossible to predict | B1 | |
## Question 8(ii):
| Answer | Marks | Notes |
|--------|-------|-------|
| Failures occur singly, unlikely as there could be a power failure that affects all lights in an area | B0B1 | |
| Failures occur independently of each other | (B1) | |
| Likely because one failure does not cause another | B1 | |
| Mean number of traffic light failures is constant each day | B0 | *OK if each hour etc* |
| Failures occur at constant average rate | (B1) | |
| Unlikely as could change with season | B0 | |
| Likely as each set has same probability of failing | B0 | |
| Likely as they run in the same mode all day | B1 | |8 In a large city the number of traffic lights that fail in one day of 24 hours is denoted by $Y$. It may be assumed that failures occur randomly.\\
(i) Explain what the statement "failures occur randomly" means.\\
(ii) State, in context, two different conditions that must be satisfied if $Y$ is to be modelled by a Poisson distribution, and for each condition explain whether you think it is likely to be met in this context.\\
(iii) For this part you may assume that $Y$ is well modelled by the distribution $\operatorname { Po } ( \lambda )$. It is given that $\mathrm { P } ( Y = 7 ) = \mathrm { P } ( Y = 8 )$. Use an algebraic method to calculate the value of $\lambda$ and hence calculate the corresponding value of $\mathrm { P } ( Y = 7 )$.
\hfill \mbox{\textit{OCR S2 2013 Q8 [10]}}