| Exam Board | OCR MEI |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2009 |
| Session | June |
| Marks | 19 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Finding minimum n for P(X≥k) threshold |
| Difficulty | Moderate -0.3 This is a comprehensive but standard Poisson distribution question covering typical S2 content: stating conditions (routine recall), basic probability calculations, scaling parameters, normal approximation, and inverse probability. Part (i) requires only memorized conditions, parts (ii)-(iv) are textbook applications, and part (v) involves straightforward inequality solving. While multi-part, each component uses well-practiced techniques without requiring novel insight. |
| 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 modelling5.02m Poisson: mean = variance = lambda5.02n Sum of Poisson variables: is Poisson |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Meteors are seen randomly and independently | B1 | |
| There is a uniform (mean) rate of occurrence of meteor sightings | B1 | Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| (A) Either \(P(X=1) = 0.6268 - 0.2725 = 0.3543\) Or \(P(X=1) = e^{-1}\frac{1.3^1}{1!} = 0.3543\) | M1, A1 | M1 for appropriate use of tables or calculation |
| (B) Using tables: \(P(X \geq 4) = 1 - P(X \leq 3) = 1 - 0.9569 = 0.0431\) | M1, A1 | M1 for appropriate probability calculation. Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\lambda = 10 \times 1.3 = 13\) | B1 | B1 for mean |
| \(P(X=10) = e^{-13}\frac{13^{10}}{10!} = 0.0859\) | M1, A1 | M1 for calculation, A1 CAO. Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Mean no. per hour \(= 60 \times 1.3 = 78\) | B1 | B1 for Normal approx. |
| Normal approx. to the Poisson, \(X \sim N(78, 78)\) | B1 | B1 for correct parameters (SOI) |
| \(P(X \geq 100) = P\left(Z > \frac{99.5 - 78}{\sqrt{78}}\right)\) | B1 | B1 for continuity correction |
| \(= P(Z > 2.434) = 1 - \Phi(2.434)\) | M1 | M1 for correct Normal probability calculation using correct tail |
| \(= 1 - 0.9926 = 0.0074\) | A1 | A1 CAO (but FT wrong or omitted CC). Total: 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(\text{at least one}) = 1 - e^{-\lambda}\frac{\lambda^0}{0!} = 1 - e^{-\lambda} \geq 0.99\) | M1 | M1 formation of equation/inequality using \(P(X \geq 1) = 1 - P(X=0)\) with Poisson distribution |
| \(e^{-\lambda} \leq 0.01\) | A1 | A1 for correct equation/inequality |
| \(-\lambda \leq \ln 0.01\), so \(\lambda \geq 4.605\) | M1, A1 | M1 for logs, A1 for 3.54 |
| \(1.3t \geq 4.605\), so \(t \geq 3.54\); Answer \(t = 4\) | A1 | A1 for \(t\) correctly justified |
| *Or* by trialling: \(t=3\): \(P = 0.9798\); \(t=4\): \(P = 0.9944\) | M1, A1, A1, A1 | M1 at least one trial; A1 correct prob; A1 both \(t=3\) and \(t=4\); A1 for \(t\). Total: 5 |
# Question 2:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Meteors are seen randomly and independently | B1 | |
| There is a uniform (mean) rate of occurrence of meteor sightings | B1 | **Total: 2** |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| (A) Either $P(X=1) = 0.6268 - 0.2725 = 0.3543$ Or $P(X=1) = e^{-1}\frac{1.3^1}{1!} = 0.3543$ | M1, A1 | M1 for appropriate use of tables or calculation |
| (B) Using tables: $P(X \geq 4) = 1 - P(X \leq 3) = 1 - 0.9569 = 0.0431$ | M1, A1 | M1 for appropriate probability calculation. **Total: 4** |
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\lambda = 10 \times 1.3 = 13$ | B1 | B1 for mean |
| $P(X=10) = e^{-13}\frac{13^{10}}{10!} = 0.0859$ | M1, A1 | M1 for calculation, A1 CAO. **Total: 3** |
## Part (iv)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Mean no. per hour $= 60 \times 1.3 = 78$ | B1 | B1 for Normal approx. |
| Normal approx. to the Poisson, $X \sim N(78, 78)$ | B1 | B1 for correct parameters (SOI) |
| $P(X \geq 100) = P\left(Z > \frac{99.5 - 78}{\sqrt{78}}\right)$ | B1 | B1 for continuity correction |
| $= P(Z > 2.434) = 1 - \Phi(2.434)$ | M1 | M1 for correct Normal probability calculation using correct tail |
| $= 1 - 0.9926 = 0.0074$ | A1 | A1 CAO (but FT wrong or omitted CC). **Total: 5** |
## Part (v)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(\text{at least one}) = 1 - e^{-\lambda}\frac{\lambda^0}{0!} = 1 - e^{-\lambda} \geq 0.99$ | M1 | M1 formation of equation/inequality using $P(X \geq 1) = 1 - P(X=0)$ with Poisson distribution |
| $e^{-\lambda} \leq 0.01$ | A1 | A1 for correct equation/inequality |
| $-\lambda \leq \ln 0.01$, so $\lambda \geq 4.605$ | M1, A1 | M1 for logs, A1 for 3.54 |
| $1.3t \geq 4.605$, so $t \geq 3.54$; Answer $t = 4$ | A1 | A1 for $t$ correctly justified |
| *Or* by trialling: $t=3$: $P = 0.9798$; $t=4$: $P = 0.9944$ | M1, A1, A1, A1 | M1 at least one trial; A1 correct prob; A1 both $t=3$ and $t=4$; A1 for $t$. **Total: 5** |
---2 Jess is watching a shower of meteors (shooting stars). During the shower, she sees meteors at an average rate of 1.3 per minute.
\begin{enumerate}[label=(\roman*)]
\item State conditions required for a Poisson distribution to be a suitable model for the number of meteors which Jess sees during a randomly selected minute.
You may assume that these conditions are satisfied.
\item Find the probability that, during one minute, Jess sees\\
(A) exactly one meteor,\\
(B) at least 4 meteors.
\item Find the probability that, in a period of 10 minutes, Jess sees exactly 10 meteors.
\item Use a suitable approximating distribution to find the probability that Jess sees a total of at least 100 meteors during a period of one hour.
\item Jess watches the shower for $t$ minutes. She wishes to be at least $99 \%$ certain that she will see one or more meteors. Find the smallest possible integer value of $t$.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI S2 2009 Q2 [19]}}