EXAMINATION SOLUTIONS
STATISTICS 1
MATH 11400
(Paper Code MATH-11400)
May/June 2012, 1 hour 30 minutes
Any general comments on the examination should be written here.
A1 (Ideas themselves are standard, dataset is unseen).
(a) (2 marks) The median is the m
UNIVERSITY OF BRISTOL
Examination for the Degrees of BBC. and M.Sci. (Level 0/4)
STATISTICS 1
MATH 11400
{Paper Code MATH-11400)
May/June 2013, 1 hour 30 minutes
This paper contains two sections, Section A and Section B.
Answer each section in a separate
Mathematics Examination Feedback Form
This form is intended to provide generic feedback to students on examination
performance in individual units, in line with university code of practice for the
assessment of taught programmes. Its purpose is to help st
EXAMINATION SOLUTIONS
STATISTICS 1
MATH 11400
(Paper Code MATH-11400)
May/June 2013, 1 hour 30 minutes
Any general comments on the examination should be written here.
A1 (Ideas themselves are standard, dataset is unseen).
Creating a R vector lebron = c(20
Mathematics Examination Feedback Form
This form is intended to provide generic feedback to students on examination
performance in individual units, in line with university code of practice for the
assessment of taught programmes. Its purpose is to help st
UNIVERSITY OF BRISTOL
Examination for the Degrees of B.Sc. and M.Sci. (Level 1)
STATISTICS 1
MATH 11400
(Paper Code MATH-11400)
May/June 2011, 1 hour 30 minutes
This paper contains two sections, Section A and Section B.
Answer each section in a separate a
Statistics 1 examination 2011 feedback on performance
Common errors, misunderstandings or other areas requiring improvement
Very few students bothered to explain their working at all (although this was not very heavily
penalised in marking); many have ver
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Further Topics in Analysis: Supplementary Exercises
1. Let F = cfw_f : (0, 1) R be the set of all functions from the segment (0, 1) to R. Prove
that card(0, 1) < card(F).
Hint: Consider the set of characteristic functions
fA (
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Further Topics in Analysis: Solutions to Supplementary Exercises
1. Let F = cfw_f : (0, 1) R be the set of all functions from the segment (0, 1) to R. Prove
that card(0, 1) < card(F).
Hint: Consider the set of characteristic f
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Mini-Revision Sheet
1. What does it mean for a set A to be countable? Is the set of all rationals with prime
denominator countable? Justify your answer.
2. What does it mean for the cardinality of a set X to be greater than or
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Further Topics in Analysis: Solution to Sample Exam Question
B. Let A R and (fn (x)nN be a sequence of functions from A to R.
(a) (6 marks)
i) What does it mean to say that the sequence (fn (x)nN converges pointwise
on A to a
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Mini-Revision Sheet: Solutions
1. What does it mean for a set A to be countable? Is the set of all rationals with prime
denominator countable? Justify your answer.
A set A is said to be countable if and only if there is a bije
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Further Topics in Analysis: Sample Exam Question
B. Let A R and (fn (x)nN be a sequence of functions from A to R.
(a) (6 marks)
i) What does it mean to say that the sequence (fn (x)nN converges pointwise
on A to a function f :
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Further Topics in Analysis: Solutions 10
1. Show that if a sequence (gn )nN of Riemann-integrable functions gn : [0, 1] R converges uniformly to a function g : [0, 1] R, then g is Riemann integrable.
Since gn g uniformly, give
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Further Topics in Analysis: Solutions 4
1. Let X and Y be sets.
(a) Prove that P(X Y ) = P(X) P(Y ).
(b) Give an example to show that P(X Y ) is not always the same as P(X) P(Y ).
(c) Give an example where X = Y and P(X Y ) =
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Further Topics in Analysis: Solutions 9
1. Let f : [0, 1] R be dened by f (x) = x. Prove that f is Riemann integrable and
compute
1
f (x) dx
0
as the limit of upper (and lower) sums.
Solution. Let Pn be the uniform partition o
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Further Topics in Analysis: Solutions 11
1. Let f : [0, 1] R be a continuous function satisfying
x
1
f (t) dt =
0
for all x [0, 1].
f (t) dt
x
Show that f (x) = 0 for all x [0, 1].
x
Dene F (x) = 0 f (t) dt for every x [0, 1].
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Further Topics in Analysis: Solutions 8
1. Find the pointwise limit of the following sequences of functions on the segment [0, 2].
Is this convergence uniform on [0, 2]?
x
(a) fn (x) = n ;
(b) fn (x) =
(c) fn (x) =
x
nx+1 ;
xn
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Further Topics in Analysis: Solutions 7
1. (a) Prove straight from the denition (that is, without appealing to Cantors theorem
on uniform continuity) that the function f (x) = x3 is uniformly continuous on
[0, 1].
(b) Prove th
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Further Topics in Analysis: Solutions 2
We assume that all sets in questions 2 and 3 are disjoint.
1. Prove that the following sets are countable.
a) The set cfw_2, 4, 6, 8, 10, . . . of positive even numbers.
b) The set cfw_
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Further Topics in Analysis: Solutions 6
1. Let (an )nN be a sequence in (0, ). Set
l := lim sup a1/n .
n
n
(i) Show that if 0
n=1 an
l < 1, then
n=1 an
(ii) Show that if l > 1, then
converges.
diverges.
(iii) Find [0, ) for wh
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Further Topics in Analysis: Solutions 3
1. Prove that the set of all polynomials with rational coecients is countable. [Hint: First,
prove that the set of polynomials of degree n is countable. Then use Lemma 3.18.]
Solution. T
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Further Topics in Analysis: Solutions 5
1. Let (an )nN be any sequence of real numbers. Which of the following must be true?
Give counterexamples for those that may be false.
(a) (an )nN has a convergent subsequence.
(b) If (a
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Further Topics in Analysis: Exercises 8
1. Find the pointwise limit of the following sequences of functions on the segment [0, 2].
Is this convergence uniform on [0, 2]?
x
(a) fn (x) = n ;
(b) fn (x) =
x
nx+1 ;
(c) fn (x) =
xn
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Further Topics in Analysis: Exercises 9
1. Let f : [0, 1] R be dened by f (x) = x. Prove that f is Riemann integrable and
compute
1
f (x) dx
0
as the limit of upper (and lower) sums.
2. Prove that the function f (x) =
x is Rie
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Further Topics in Analysis: Exercises 11
1. Let f : [0, 1] R be a continuous function satisfying
1
x
0
for all x [0, 1].
f (t) dt
f (t) dt =
x
Show that f (x) = 0 for all x [0, 1].
2. Let f : R R be a continuous function, and