OCR MEI Further Statistics Minor Specimen — Question 7 4 marks

Exam BoardOCR MEI
ModuleFurther Statistics Minor (Further Statistics Minor)
SessionSpecimen
Marks4
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Mark schemeDownload PDF ↗
TopicDiscrete Random Variables
TypeLinear combinations of independent variables
DifficultyModerate -0.5 This is a straightforward variance calculation requiring recognition that Y is the sum of 100 independent identically distributed random variables, each with variance 1. The solution involves basic variance properties (Var(aX)=a²Var(X) and additivity for independent variables) with no conceptual difficulty or novel insight required. The 4 marks reflect showing working rather than problem complexity.
Spec5.02b Expectation and variance: discrete random variables5.02d Binomial: mean np and variance np(1-p)
A fair coin has \(+1\) written on the heads side and \(-1\) on the tails side. The coin is tossed \(100\) times. The sum of the numbers showing on the \(100\) tosses is the random variable \(Y\). Show that the variance of \(Y\) is \(100\). [4]
Question 7:
AnswerMarks
7Let X be the number of heads on the 100 tosses
[X ~ Bin (100, 0.5)]
Y (cid:32)X (cid:16)(100(cid:16)X)(cid:32)2X (cid:16)100
Var (Y) = 4 Var (X)
Var (X) = npq so Var(Y)(cid:32)4(cid:117)100(cid:117)0.5(cid:117)0.5
AnswerMarks
= 100 AGM1
M1
IM1
E1
AnswerMarks
[4]M3.3
3.1b
1.1
AnswerMarks Guidance
2.1E
QuestionAO1 AO2
1i1 0
1ii1 0
1iii1 0
2iA0 1
2iB3 0
2ii2 0
2iii0 1
2iv1 0
3i2 0
3ii2 0
3iii1 1
4i0 0
4iiA1 0
1
AnswerMarks Guidance
4iiB1 0
4iiiA0 0
11
4iiiB1 0
5i0 1
5ii3 0
00 3
5iii1 2
5iv0 0
01 1
6i2 0
6ii0 C
10 0
6iii0 1
6iv2 1
6vE
31 0
6vi1 1
7P
11 1
Totals30 12
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Question 7:
7 | Let X be the number of heads on the 100 tosses
[X ~ Bin (100, 0.5)]
Y (cid:32)X (cid:16)(100(cid:16)X)(cid:32)2X (cid:16)100
Var (Y) = 4 Var (X)
Var (X) = npq so Var(Y)(cid:32)4(cid:117)100(cid:117)0.5(cid:117)0.5
= 100 AG | M1
M1
IM1
E1
[4] | M3.3
3.1b
1.1
2.1 | E
Question | AO1 | AO2 | AO3(PS) | AO3(M) | Totals
1i | 1 | 0 | 0 | 1 | 2
1ii | 1 | 0 | 0 | 0 | 1
1iii | 1 | 0 | 0 | 0 | 1
2iA | 0 | 1 | 0 | 0 | 1
2iB | 3 | 0 | 0 | 0 | 3
2ii | 2 | 0 | 0 | 0 | 2
2iii | 0 | 1 | 0 | 0 | 1
2iv | 1 | 0 | 0 | 0 | 1
3i | 2 | 0 | 0 | 1 | 3
3ii | 2 | 0 | 0 | 1 | 3
3iii | 1 | 1 | 2 | 0 | 4
4i | 0 | 0 | 0 | 2 | 2
4iiA | 1 | 0 | 0 | 0 | N
1
4iiB | 1 | 0 | 0 | 1 | 2
4iiiA | 0 | 0 | 0 | E
1 | 1
4iiiB | 1 | 0 | 0 | 1 | 2
5i | 0 | 1 | 0 | 0 | 1
5ii | 3 | 0 | M
0 | 0 | 3
5iii | 1 | 2 | 0 | 2 | 5
5iv | 0 | 0 | I
0 | 1 | 1
6i | 2 | 0 | 0 | 0 | 2
6ii | 0 | C
1 | 0 | 0 | 1
6iii | 0 | 1 | 0 | 0 | 1
6iv | 2 | 1 | 0 | 1 | 4
6v | E
3 | 1 | 0 | 1 | 5
6vi | 1 | 1 | 1 | 0 | 3
7 | P
1 | 1 | 1 | 1 | 4
Totals | 30 | 12 | 4 | 14 | 60
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A fair coin has $+1$ written on the heads side and $-1$ on the tails side. The coin is tossed $100$ times. The sum of the numbers showing on the $100$ tosses is the random variable $Y$. Show that the variance of $Y$ is $100$. [4]

\hfill \mbox{\textit{OCR MEI Further Statistics Minor  Q7 [4]}}
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