CAIE S2 2016 March — Question 1 5 marks

Exam BoardCAIE
ModuleS2 (Statistics 2)
Year2016
SessionMarch
Marks5
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TopicLinear combinations of normal random variables
TypePure expectation and variance calculation
DifficultyModerate -0.8 This is a straightforward application of binomial distribution properties and linear combinations. Students need only identify X ~ B(20, 1/6) and Y ~ B(20, 1/2), then apply E(X+Y) = E(X) + E(Y) and Var(X+Y) = Var(X) + Var(Y) (using independence). No problem-solving or insight required—pure routine calculation.
Spec2.04b Binomial distribution: as model B(n,p)5.04a Linear combinations: E(aX+bY), Var(aX+bY)
1 A fair six-sided die is thrown 20 times and the number of sixes, \(X\), is recorded. Another fair six-sided die is thrown 20 times and the number of odd-numbered scores, \(Y\), is recorded. Find the mean and standard deviation of \(X + Y\).
Question 1:
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(E(X) = \frac{10}{3}\) oe, \(\text{Var}(X) = \frac{25}{9}\) oeB1 For \(E(X)\) and \(\text{Var}(X)\)
\(E(Y) = 10\), \(\text{Var}(Y) = 5\)B1 For \(E(Y)\) and \(\text{Var}(Y)\); OR for \(E(X)\) and \(E(Y)\); for \(\text{Var}(X)\) and \(\text{Var}(Y)\)
\(E(X+Y) = \frac{40}{3}\) oe or 13.3 (3 sf)B1
\(\text{Var}(X+Y) = \text{"}\frac{25}{9}\text{"} + \text{"5"}\)M1 For adding 2 (appropriate) variances
\(\text{sd} = \frac{\sqrt{70}}{3}\) oe or 2.79 (3 sf)A1 [5] or sd \(= \sqrt{2} \times \frac{5}{3}\) 
## Question 1:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $E(X) = \frac{10}{3}$ oe, $\text{Var}(X) = \frac{25}{9}$ oe | **B1** | For $E(X)$ and $\text{Var}(X)$ |
| $E(Y) = 10$, $\text{Var}(Y) = 5$ | **B1** | For $E(Y)$ and $\text{Var}(Y)$; OR for $E(X)$ and $E(Y)$; for $\text{Var}(X)$ and $\text{Var}(Y)$ |
| $E(X+Y) = \frac{40}{3}$ oe or 13.3 (3 sf) | **B1** | |
| $\text{Var}(X+Y) = \text{"}\frac{25}{9}\text{"} + \text{"5"}$ | **M1** | For adding 2 (appropriate) variances |
| $\text{sd} = \frac{\sqrt{70}}{3}$ oe or 2.79 (3 sf) | **A1** [5] | or sd $= \sqrt{2} \times \frac{5}{3}$ |

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1 A fair six-sided die is thrown 20 times and the number of sixes, $X$, is recorded. Another fair six-sided die is thrown 20 times and the number of odd-numbered scores, $Y$, is recorded. Find the mean and standard deviation of $X + Y$.

\hfill \mbox{\textit{CAIE S2 2016 Q1 [5]}}
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