| Exam Board | OCR MEI |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2010 |
| Session | June |
| Marks | 19 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Two independent Poisson sums |
| Difficulty | Moderate -0.8 This is a straightforward application of standard Poisson distribution properties and calculations. Part (i) requires recall of textbook conditions, part (ii) is immediate recall that variance equals mean, and remaining parts involve routine Poisson probability calculations and normal approximation—all standard S2 techniques with no problem-solving insight required. |
| 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 |
|---|---|---|
| Part (i) | Counts have a uniform average rate of occurrence and all counts are independent | E1 |
| Part (ii) | Variance = 3.4 | B1 |
| Part (iii) | (A) Either \(P(X = 3) = 0.5584 - 0.3397 = 0.2187\) or \(P(X = 3) = e^{-3.4}\frac{3.4^3}{3!} = 0.2186\) | M1 for use of tables or calculation; A1 |
| (B) Using tables: \(P(X \geq 3) = 1 - P(X \leq 2) = 1 - 0.3397 = 0.6603\) | M1 for \(1 - P(X \leq 2)\); M1 correct use of Poisson tables; A1 | 3 |
| Part (iv) | \(\lambda = 12 \times 3.4 = 40.8\) | B1 for mean |
| \(P(X = 40) = e^{-40.8}\frac{40.8^{40}}{40!} = 0.0625\) | M1 for calculation; A1 | 3 |
| Part (v) | Mean no. per hour = \(12 \times 3.4 = 40.8\) | B1 for Normal approx.; B1 for correct parameters (SOI) |
| Using Normal approx. to the Poisson, \(X \sim N(40.8, 40.8)\) | ||
| \(P(X \geq 40) = P\left(Z \geq \frac{39.5 - 40.8}{\sqrt{40.8}}\right)\) | B1 for correct continuity corr. | |
| \(= P(Z > -0.2035) = \Phi(0.2035) = 0.5806\) | M1 for probability using correct tail; A1 CAO (3 s.f.) | 5 |
| Part (vi) | Overall mean = 4.8 | B1 for 4.8 |
| \(P(X \geq 8) = 1 - P(X \leq 7) = 1 - 0.8867 = 0.1133\) | M1; A1 | 3 |
| TOTAL | 19 |
| **Part (i)** | Counts have a uniform average rate of occurrence and all counts are independent | E1 | 2 |
|---|---|---|---|
| **Part (ii)** | Variance = 3.4 | B1 | 1 |
| **Part (iii)** | **(A)** Either $P(X = 3) = 0.5584 - 0.3397 = 0.2187$ or $P(X = 3) = e^{-3.4}\frac{3.4^3}{3!} = 0.2186$ | M1 for use of tables or calculation; A1 | 2 |
| | **(B)** Using tables: $P(X \geq 3) = 1 - P(X \leq 2) = 1 - 0.3397 = 0.6603$ | M1 for $1 - P(X \leq 2)$; M1 correct use of Poisson tables; A1 | 3 |
| **Part (iv)** | $\lambda = 12 \times 3.4 = 40.8$ | B1 for mean | |
| | $P(X = 40) = e^{-40.8}\frac{40.8^{40}}{40!} = 0.0625$ | M1 for calculation; A1 | 3 |
| **Part (v)** | Mean no. per hour = $12 \times 3.4 = 40.8$ | B1 for Normal approx.; B1 for correct parameters (SOI) | |
| | Using Normal approx. to the Poisson, $X \sim N(40.8, 40.8)$ | | |
| | $P(X \geq 40) = P\left(Z \geq \frac{39.5 - 40.8}{\sqrt{40.8}}\right)$ | B1 for correct continuity corr. | |
| | $= P(Z > -0.2035) = \Phi(0.2035) = 0.5806$ | M1 for probability using correct tail; A1 CAO (3 s.f.) | 5 |
| **Part (vi)** | Overall mean = 4.8 | B1 for 4.8 | |
| | $P(X \geq 8) = 1 - P(X \leq 7) = 1 - 0.8867 = 0.1133$ | M1; A1 | 3 |
| | | **TOTAL** | **19** |
---2 A radioactive source is decaying at a mean rate of 3.4 counts per 5 seconds.
\begin{enumerate}[label=(\roman*)]
\item State conditions for a Poisson distribution to be a suitable model for the rate of decay of the source.
You may assume that a Poisson distribution with a mean rate of 3.4 counts per 5 seconds is a suitable model.
\item State the variance of this Poisson distribution.
\item Find the probability of\\
(A) exactly 3 counts in a 5 -second period,\\
(B) at least 3 counts in a 5 -second period.
\item Find the probability of exactly 40 counts in a period of 60 seconds.
\item Use a suitable approximating distribution to find the probability of at least 40 counts in a period of 60 seconds.
\item The background radiation rate also, independently, follows a Poisson distribution and produces a mean count of 1.4 per 5 seconds. Find the probability that the radiation source together with the background radiation give a total count of at least 8 in a 5 -second period.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI S2 2010 Q2 [19]}}