CAIE S2 2004 November — Question 1 4 marks

Exam BoardCAIE
ModuleS2 (Statistics 2)
Year2004
SessionNovember
Marks4
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Mark schemeDownload PDF ↗
TopicSum of Poisson processes
TypeBasic sum of two Poissons
DifficultyStandard +0.3 This is a straightforward application of the Poisson distribution property that the sum of independent Poisson variables is also Poisson. Students need to recognize that the combined rate is 1.7 + 0.6 = 2.3 per second, scale to 3 seconds (λ = 6.9), then calculate P(X=6) + P(X=7) + P(X=8) using the standard formula. While it requires understanding of Poisson properties and careful arithmetic, it's a routine textbook-style question with no novel insight required.
Spec5.02n Sum of Poisson variables: is Poisson
1 The number of radioactive particles emitted per second by a certain metal is random and has mean 1.7. The radioactive metal is placed next to an object which independently emits particles at random such that the mean number of particles emitted per second is 0.6 . Find the probability that the total number of particles emitted in the next 3 seconds is 6, 7 or 8 .
Question 1:
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(\lambda = 2.3 \times 3 = 6.9\)M1 For attempt at Poisson, any mean
\(P(6,7,8) = e^{-6.9}\left(\frac{6.9^6}{6!} + \frac{6.9^7}{7!} + \frac{6.9^8}{8!}\right)\)A1 For correct mean
\(= e^{-6.9}(425.06)\)A1ft For correct expression with their mean
\(= 0.428\)A1 4 For correct answer 
## Question 1:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\lambda = 2.3 \times 3 = 6.9$ | M1 | For attempt at Poisson, any mean |
| $P(6,7,8) = e^{-6.9}\left(\frac{6.9^6}{6!} + \frac{6.9^7}{7!} + \frac{6.9^8}{8!}\right)$ | A1 | For correct mean |
| $= e^{-6.9}(425.06)$ | A1ft | For correct expression with their mean |
| $= 0.428$ | A1 **4** | For correct answer |

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1 The number of radioactive particles emitted per second by a certain metal is random and has mean 1.7. The radioactive metal is placed next to an object which independently emits particles at random such that the mean number of particles emitted per second is 0.6 . Find the probability that the total number of particles emitted in the next 3 seconds is 6, 7 or 8 .

\hfill \mbox{\textit{CAIE S2 2004 Q1 [4]}}
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