OCR Further Statistics 2021 November — Question 8 11 marks

Exam BoardOCR
ModuleFurther Statistics (Further Statistics)
Year2021
SessionNovember
Marks11
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicContinuous Uniform Random Variables
TypeTranscendental function expectations
DifficultyChallenging +1.8 This Further Maths question requires computing E[a cos(aY)] by integration, solving a transcendental equation numerically (sin(2a)/(2a) = 0.3/a), and finding percentiles of transformed variables. The integration by parts and transcendental equation solving elevate this beyond standard A-level, though the uniform distribution setup is straightforward.
Spec5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration5.03f Relate pdf-cdf: medians and percentiles
8 The continuous random variable \(Y\) has a uniform distribution on [0,2].
  1. It is given that \(\mathrm { E } [ a \cos ( a Y ) ] = 0.3\), where \(a\) is a constant between 0 and 1 , and \(a Y\) is measured in radians. Determine the value of the constant \(a\).
  2. Determine the \(60 ^ { \text {th } }\) percentile of \(Y ^ { 2 }\).
Question 8:
AnswerMarks Guidance
8(a) f(x) = ½
2
∫ 1acos(ax)dx= 0.3
0 2
2
1sin(ax)
 
2 0
1sin(2a)=0.3
2
AnswerMarks
a = 0.32175…B1
M1
B1
M1
A1
AnswerMarks
[5]3.3
3.1a
1.1
2.1
AnswerMarks
1.1Stated or implied, e.g. on diagram
∫f(x) a cos ax dx & equated to 0.3
Correct indefinite integral
Correct limits, solve
Answer, a.r.t. 0.322 (ignore other answers)
AnswerMarks Guidance
8(b) F(y) = ½ y [0 ≤ y ≤ 2]
P(Y 2 ≤ m) = P(0 < Y ≤ √m)
= F(√m) [= ½√m]
½√P = 0.6
60
P = 1.44
AnswerMarks
60M1
A1
M1
A1
M1
A1
AnswerMarks
[6]3.1a
1.1
2.1
1.1
1.1
AnswerMarks
2.2aUse their f(y) to obtain CDF
Correct F(y) (range need not be stated explicitly)
Find CDF of Y 2, allow m2 instead of √m , or ±√m, here
Use F(y) correctly
Equate to 0.6 and solve, need √m here
1.44 or exact equivalent
PMT
OCR (Oxford Cambridge and RSA Examinations)
The Triangle Building
Shaftesbury Road
Cambridge
CB2 8EA
OCR Customer Contact Centre
Education and Learning
Telephone: 01223 553998
Facsimile: 01223 552627
Email: general.qualifications@ocr.org.uk
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For staff training purposes and as part of our quality assurance programme your call may be
recorded or monitored
Question 8:
8 | (a) | f(x) = ½
2
∫ 1acos(ax)dx= 0.3
0 2
2
1sin(ax)
 
2 0
1sin(2a)=0.3
2
a = 0.32175… | B1
M1
B1
M1
A1
[5] | 3.3
3.1a
1.1
2.1
1.1 | Stated or implied, e.g. on diagram
∫f(x) a cos ax dx & equated to 0.3
Correct indefinite integral
Correct limits, solve
Answer, a.r.t. 0.322 (ignore other answers)
8 | (b) | F(y) = ½ y [0 ≤ y ≤ 2]
P(Y 2 ≤ m) = P(0 < Y ≤ √m)
= F(√m) [= ½√m]
½√P = 0.6
60
P = 1.44
60 | M1
A1
M1
A1
M1
A1
[6] | 3.1a
1.1
2.1
1.1
1.1
2.2a | Use their f(y) to obtain CDF
Correct F(y) (range need not be stated explicitly)
Find CDF of Y 2, allow m2 instead of √m , or ±√m, here
Use F(y) correctly
Equate to 0.6 and solve, need √m here
1.44 or exact equivalent
PMT
OCR (Oxford Cambridge and RSA Examinations)
The Triangle Building
Shaftesbury Road
Cambridge
CB2 8EA
OCR Customer Contact Centre
Education and Learning
Telephone: 01223 553998
Facsimile: 01223 552627
Email: general.qualifications@ocr.org.uk
www.ocr.org.uk
For staff training purposes and as part of our quality assurance programme your call may be
recorded or monitored
8 The continuous random variable $Y$ has a uniform distribution on [0,2].
\begin{enumerate}[label=(\alph*)]
\item It is given that $\mathrm { E } [ a \cos ( a Y ) ] = 0.3$, where $a$ is a constant between 0 and 1 , and $a Y$ is measured in radians.

Determine the value of the constant $a$.
\item Determine the $60 ^ { \text {th } }$ percentile of $Y ^ { 2 }$.
\end{enumerate}

\hfill \mbox{\textit{OCR Further Statistics 2021 Q8 [11]}}
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