Question 1 - A-Level Maths

CAIE S2 2021 June — Question 1 5 marks

Exam Board	CAIE
Module	S2 (Statistics 2)
Year	2021
Session	June
Marks	5
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Poisson distribution
Type	Expectation and variance of Poisson-related expressions
Difficulty	Moderate -0.3 This is a straightforward application of Poisson distribution properties with linear transformation. Students need to recognize that total goals in 10 matches follows Poisson(13.6), then apply E(5X)=5E(X) and SD(5X)=5SD(X). The calculation is routine with no conceptual challenges beyond basic distribution properties.
Spec	5.02i Poisson distribution: random events model 5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities 5.02m Poisson: mean = variance = lambda

1 The number of goals scored by a team in a match is independent of other matches, and is denoted by the random variable $X$, which has a Poisson distribution with mean 1.36. A supporter offers to make a donation of $\$ 5$ to the team for each goal that they score in the next 10 matches. Find the expectation and standard deviation of the amount that the supporter will pay.

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Question 1:

Answer	Marks	Guidance
Answer	Marks	Guidance
$\lambda = 10 \times 1.36 = 13.6$	M1
$E(\text{amount}) = 5 \times 13.6 = \$68$	A1
$\text{Var}(\text{amount}) = 5^2 \times 13.6 = 340$	M1	$5^2 \times \ldots$
M1	$\ldots \times \text{their } \lambda$
$\text{Standard Deviation} = \$18.4 \text{ (3 s.f.)}\(	A1	CAO, condone \)2\sqrt{85}$
Total: 5

## Question 1:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\lambda = 10 \times 1.36 = 13.6$ | M1 | |
| $E(\text{amount}) = 5 \times 13.6 = \$68$ | A1 | |
| $\text{Var}(\text{amount}) = 5^2 \times 13.6 = 340$ | M1 | $5^2 \times \ldots$ |
| | M1 | $\ldots \times \text{their } \lambda$ |
| $\text{Standard Deviation} = \$18.4 \text{ (3 s.f.)}$ | A1 | CAO, condone $2\sqrt{85}$ |
| | **Total: 5** | |

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1 The number of goals scored by a team in a match is independent of other matches, and is denoted by the random variable $X$, which has a Poisson distribution with mean 1.36. A supporter offers to make a donation of $\$ 5$ to the team for each goal that they score in the next 10 matches.

Find the expectation and standard deviation of the amount that the supporter will pay.\\

\hfill \mbox{\textit{CAIE S2 2021 Q1 [5]}}

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Q1 5 Q2 8 Q3 6 Q4 9 Q5 9 Q6 13