| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2010 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sum of Poisson processes |
| Type | Multi-period repeated application |
| Difficulty | Moderate -0.3 This is a straightforward application of Poisson distribution with clearly stated rate parameter. Parts (a)-(c) require direct substitution into standard formulas (P(X=0) for different time periods), while part (d) tests understanding of the memoryless property. The question is slightly easier than average because it's entirely procedural with no problem-solving or proof required, though it does test conceptual understanding in part (d). |
| Spec | 5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(Y \sim \text{Po}(0.25)\) | B1 | \(P(Y=0) = e^{-0.25} = 0.7788\) |
| (b) \(X \sim \text{Po}(0.4)\) | B1 | \(P(\text{Robot will break down}) = 1 - P(X = 0) = 1 - e^{-0.4} = 1 - 0.067032 = 0.3297\) |
| (c) \(P(X = 2) = \frac{e^{-0.4}(0.4)^2}{2} = 0.0536\) | M1 A1 | (2 marks) |
| (d) \(0.3297\) or answer to part (b) as Poisson events are independent | B1ft B1 dep | (2 marks) |
| Total [10] |
**(a)** $Y \sim \text{Po}(0.25)$ | B1 | $P(Y=0) = e^{-0.25} = 0.7788$ | M1 A1 | (3 marks)
**(b)** $X \sim \text{Po}(0.4)$ | B1 | $P(\text{Robot will break down}) = 1 - P(X = 0) = 1 - e^{-0.4} = 1 - 0.067032 = 0.3297$ | M1 A1 | (3 marks)
**(c)** $P(X = 2) = \frac{e^{-0.4}(0.4)^2}{2} = 0.0536$ | M1 A1 | (2 marks)
**(d)** $0.3297$ or answer to part (b) as Poisson events are **independent** | B1ft B1 dep | (2 marks)
| **Total [10]**
---\begin{enumerate}
\item A robot is programmed to build cars on a production line. The robot breaks down at random at a rate of once every 20 hours.\\
(a) Find the probability that it will work continuously for 5 hours without a breakdown.
\end{enumerate}
Find the probability that, in an 8 hour period,\\
(b) the robot will break down at least once,\\
(c) there are exactly 2 breakdowns.
In a particular 8 hour period, the robot broke down twice.\\
(d) Write down the probability that the robot will break down in the following 8 hour period. Give a reason for your answer.\\
\hfill \mbox{\textit{Edexcel S2 2010 Q3 [10]}}