CAIE S2 2011 June — Question 1 4 marks

Exam BoardCAIE
ModuleS2 (Statistics 2)
Year2011
SessionJune
Marks4
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TopicSum of Poisson processes
TypeBasic sum of two Poissons
DifficultyModerate -0.3 This question requires knowing that independent Poisson distributions sum to another Poisson distribution and that the mean scales linearly with time. The calculation itself is straightforward: find P(X < 3) for a Poisson with mean 3.4. While it tests understanding of two key Poisson properties, the execution is routine once the setup is recognized, making it slightly easier than average.
Spec5.02i 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
1 A hotel kitchen has two dish-washing machines. The numbers of breakdowns per year by the two machines have independent Poisson distributions with means 0.7 and 1.0 . Find the probability that the total number of breakdowns by the two machines during the next two years will be less than 3 .
Question 1:
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\((0.7 + 1.0) \times 2 = 3.4\)M1, A1 Attempt combined mean
\(e^{-3.4}(1 + 3.4 + \frac{3.4^2}{2})= 0.34(0)\)M1, A1 Poisson P(0,1,2), any \(\lambda\) (allow one end error)
Alternative: By CombinationsM2, A1, A1[4] At least 4 correct \(\lambda=1.4\), \(\lambda=2\); All 6 correct combinations; Correct answer 
## Question 1:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $(0.7 + 1.0) \times 2 = 3.4$ | M1, A1 | Attempt combined mean |
| $e^{-3.4}(1 + 3.4 + \frac{3.4^2}{2})= 0.34(0)$ | M1, A1 | Poisson P(0,1,2), any $\lambda$ (allow one end error) |
| **Alternative:** By Combinations | M2, A1, A1[4] | At least 4 correct $\lambda=1.4$, $\lambda=2$; All 6 correct combinations; Correct answer |

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1 A hotel kitchen has two dish-washing machines. The numbers of breakdowns per year by the two machines have independent Poisson distributions with means 0.7 and 1.0 . Find the probability that the total number of breakdowns by the two machines during the next two years will be less than 3 .

\hfill \mbox{\textit{CAIE S2 2011 Q1 [4]}}
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