CAIE S2 2012 June — Question 2 5 marks

Exam BoardCAIE
ModuleS2 (Statistics 2)
Year2012
SessionJune
Marks5
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TopicLinear combinations of normal random variables
TypeTwo or more different variables
DifficultyStandard +0.3 This is a straightforward application of linear combinations of independent normal random variables. Students need to find the mean and variance of 3X - Y using standard formulas (E[aX + bY] and Var[aX + bY] for independent variables), then standardize and use normal tables. It's slightly above average difficulty due to requiring multiple steps and careful arithmetic with the coefficients, but it's a standard textbook exercise with no novel insight required.
Spec5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.04b Linear combinations: of normal distributions
2 The independent random variables \(X\) and \(Y\) have the distributions \(\mathrm { N } ( 6.5,14 )\) and \(\mathrm { N } ( 7.4,15 )\) respectively. Find \(\mathrm { P } ( 3 X - Y < 20 )\).
Question 2:
Part (i)
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(E(3X - Y) = 12.1\)B1
\(\text{Var}(3X - Y) = 9 \times 14 + 15 = 141\)B1
\(\frac{20 - 12.1}{\sqrt{141}} = 0.665\)M1 Allow without \(\sqrt{}\) (No Continuity Correction)
\(\Phi(0.665)\)M1 Correct area consistent with their working
\(= 0.747\) (3 sf)A1 (5) 
## Question 2:

### Part (i)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $E(3X - Y) = 12.1$ | B1 | |
| $\text{Var}(3X - Y) = 9 \times 14 + 15 = 141$ | B1 | |
| $\frac{20 - 12.1}{\sqrt{141}} = 0.665$ | M1 | Allow without $\sqrt{}$ (No Continuity Correction) |
| $\Phi(0.665)$ | M1 | Correct area consistent with their working |
| $= 0.747$ (3 sf) | A1 (5) | |

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2 The independent random variables $X$ and $Y$ have the distributions $\mathrm { N } ( 6.5,14 )$ and $\mathrm { N } ( 7.4,15 )$ respectively. Find $\mathrm { P } ( 3 X - Y < 20 )$.

\hfill \mbox{\textit{CAIE S2 2012 Q2 [5]}}
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