Question 3 - A-Level Maths

CAIE S2 2015 June — Question 3 5 marks

Exam Board	CAIE
Module	S2 (Statistics 2)
Year	2015
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 payment over 5 days follows Y = 200X where X ~ Po(5 × 0.15), then apply E(aX) = aE(X) and Var(aX) = a²Var(X). The calculation is routine once the setup is identified, making it slightly easier than average.
Spec	5.02i Poisson distribution: random events model 5.02m Poisson: mean = variance = lambda 5.04a Linear combinations: E(aX+bY), Var(aX+bY)

3 In a golf tournament, the number of times in a day that a 'hole-in-one' is scored is denoted by the variable $X$, which has a Poisson distribution with mean 0.15 . Mr Crump offers to pay $\$ 200$ each time that a hole-in-one is scored during 5 days of play. Find the expectation and variance of the amount that Mr Crump pays.

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Answer	Marks
$\lambda = 5 \times 0.15$ (= 0.75)	M1
$E(\text{amount}) = 200 \times 0.75 = 150$	A1
$\text{Var}(\text{weekly no of hole-in-ones}) = 0.75$	B1
$\text{Var}(\text{amount}) = 200^2 \times 0.75 = 30,000$	M1
A1

Allow $200^2 \times$ their variance (with nothing added/subtracted at any stage)

(SR probability table can score M1A0 srB1 if var rounds to 30,000 (2sf))

Total: 5

$\lambda = 5 \times 0.15$ (= 0.75) | M1

$E(\text{amount}) = 200 \times 0.75 = 150$ | A1

$\text{Var}(\text{weekly no of hole-in-ones}) = 0.75$ | B1

$\text{Var}(\text{amount}) = 200^2 \times 0.75 = 30,000$ | M1

| A1

Allow $200^2 \times$ their variance (with nothing added/subtracted at any stage)

(SR probability table can score M1A0 srB1 if var rounds to 30,000 (2sf))

**Total: 5**

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3 In a golf tournament, the number of times in a day that a 'hole-in-one' is scored is denoted by the variable $X$, which has a Poisson distribution with mean 0.15 . Mr Crump offers to pay $\$ 200$ each time that a hole-in-one is scored during 5 days of play. Find the expectation and variance of the amount that Mr Crump pays.

\hfill \mbox{\textit{CAIE S2 2015 Q3 [5]}}

This paper (7 questions)

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Q1 3 Q2 5 Q3 5 Q4 8 Q5 9 Q6 10 Q7 10

Answer	Marks
\(\lambda = 5 \times 0.15\) (= 0.75)	M1
\(E(\text{amount}) = 200 \times 0.75 = 150\)	A1
\(\text{Var}(\text{weekly no of hole-in-ones}) = 0.75\)	B1
\(\text{Var}(\text{amount}) = 200^2 \times 0.75 = 30,000\)	M1
A1

CAIE S2 2015 June — Question 3 5 marks

Allow \(200^2 \times\) their variance (with nothing added/subtracted at any stage)

(SR probability table can score M1A0 srB1 if var rounds to 30,000 (2sf))

Total: 5

This paper (7 questions)