CAIE S2 2018 June — Question 1 3 marks

Exam BoardCAIE
ModuleS2 (Statistics 2)
Year2018
SessionJune
Marks3
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicSum of Poisson processes
TypeSum of three or more Poissons
DifficultyStandard +0.3 This question requires knowing that Poisson distributions sum (λ_total = 0.2+0.3+0.6 = 1.1) and scale with time (4 minutes gives λ = 4.4), then calculating P(X < 4) = P(X ≤ 3) using standard Poisson probability formula. It's a straightforward application of two key Poisson properties with routine calculation, 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 The numbers of alpha, beta and gamma particles emitted per minute by a certain piece of rock have independent distributions \(\operatorname { Po } ( 0.2 ) , \operatorname { Po } ( 0.3 )\) and \(\operatorname { Po } ( 0.6 )\) respectively. Find the probability that the total number of particles emitted during a 4 -minute period is less than 4.

Question 1:
AnswerMarks Guidance
\(\lambda = 4.4\)B1
\(P(X<4) = e^{-4.4}(1 + 4.4 + \frac{4.4^2}{2} + \frac{4.4^3}{3!})\)M1 Allow any \(\lambda\), allow one end error
\(= 0.359\)A1
Total: 3
**Question 1:**

$\lambda = 4.4$ | B1 |

$P(X<4) = e^{-4.4}(1 + 4.4 + \frac{4.4^2}{2} + \frac{4.4^3}{3!})$ | M1 | Allow any $\lambda$, allow one end error

$= 0.359$ | A1 |

**Total: 3**

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1 The numbers of alpha, beta and gamma particles emitted per minute by a certain piece of rock have independent distributions $\operatorname { Po } ( 0.2 ) , \operatorname { Po } ( 0.3 )$ and $\operatorname { Po } ( 0.6 )$ respectively. Find the probability that the total number of particles emitted during a 4 -minute period is less than 4.\\

\hfill \mbox{\textit{CAIE S2 2018 Q1 [3]}}