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INSTITUTE AND FACULTY OF ACTUARIES
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EXAMINATION
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19 April 2016 (am)
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Subject CT8 Financial Economics
Core Technical
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Time allowed: Three hours
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INSTRUCTIONS TO THE CANDI
PAPER CODE NO.
MATH263
EXAMINERS: Dr. A. Haddley
DEPARTMENT: Mathematical Sciences
TEL.NO. 45584
SPECIMEN EXAMINATIONS
Statistical Theory and Methods I
Time allowed: Two and a half hours
INSTRUCTIONS
Exam Style Problems - Week 7
Math263
1. A farmer has three similar tomato patches. For the last 10 years, he has treated one patch with
natural fertiliser, one with artificial fertiliser, and one with
Exam Style Problems - Week 5
Math263
1. A random sample of 10 men and 12 women swam 100 metres. Their times in seconds were recorded
as follows.
Men (X1 )
101.0
88.2
144.1
81.6
129.9
85.0
117.1
140.4
Exam Style Problems - Week 5
Math263
1. A random sample of 10 men and 12 women swam 100 metres. Their times in seconds were recorded
as follows.
Men (X1 )
101.0
88.2
144.1
81.6
129.9
85.0
117.1
140.4
Exam Style Problems - Week 7
Math263
1. A farmer has three similar tomato patches. For the last 10 years, he has treated one patch with
natural fertiliser, one with artificial fertiliser, and one with
Exam Style Problems - Week 3
Math263
1. A researcher wants to compare the abilities of two sets of students. He finds 100 mathematics students,
and 150 students with no mathematical background, and ha
MATH365 Measure Theory
Feedback on Exercise Sheet 3.
Problem 1. For a fixed irrational x [0, 1], we need to prove that
> 0 > 0 : y [0, 1] |x y| < = |f (y)| < .
1
(Note that x > 0.) For an arbitrarily
MATH365 Measure Theory
Feedback on Exercise Sheet 2.
Problem 1. We need to say that, for some positive , the statement
N N : n > N |fn A| <
(1)
is false. Therefore, the required formula begins with >
Section 3 (MATH365)
Spaces of Integrable Functions. .
Normed Spaces
Definition. A vector space (over R) is a set X which satisfies the
following axioms:
(i) if x, y X then there is uniquely defined el
MATH365 Measure Theory
Feedback on Exercise Sheet 1.
Problem 1. Let m E N. Then m does not belong to the subset Am+1 : cfw_m + 1, m + 2, . . .
and does not belong to A. Hence A contains no elements: A
Lecture Notes
for
Financial Mathematics I
Lecturers: Corina Constantinescu and Sule Sahin
University of Liverpool
Fall 2017
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Contents
1 Geometric Series
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2 Time value of money
2.1 Effective rate of
Math261 - Exercise Sheet 11
The contents of this exercise sheet will not be assessed by an Assignment, but may appear on the exam.
Q1. A 24-hour fruit shop buys individual apples for 15p each, and sel
Math261 - Exercise Sheet 9
Help for this exercise sheet will be available during the tutorial in week 10. You do not need to submit any
solutions to these questions, but it should help you to prepare
Math261 - Exercise Sheet 7
Help for this exercise sheet will be available during the tutorial in week 8. You do not need to submit any
solutions to these questions, but it should help you to prepare f
Math261 - Exercise Sheet 10
Help for this exercise sheet will be available during the tutorial in week 11. You do not need to submit any
solutions to these questions, but it should help you to prepare
Math261 - Exercise Sheet 13
There will be a revision lecture on Thursday 11th January 2018 at 9am. It is hard to know exactly what
material I should include:
I cant revise 36 hours of lectures in one
Math261 - Exercise Sheet 12
The contents of this exercise sheet will not be assessed by an Assignment, but may appear on the exam.
Q1. Consider the linear program,
Maximise: z = x1 + 2x2
subject to
3x
Math261 - Exercise Sheet 6
Help for this exercise sheet will be available during the tutorial in week 7. You do not need to submit any
solutions to these questions, but it should help you to prepare f
MATH267 Homework Sheet 10
Deadline to upload your group solutions, 9am, Monday, December 11, 2017.
1. (a) The current term structure is defined by st = 0.06+0.01t for t = 1, 2, 3. Calculate
the at-par
Math261 - Exercise Sheet 8
Help for this exercise sheet will be available during the tutorial in week 9. You do not need to submit any
solutions to these questions, but it should help you to prepare f
MATH263: Statistical Theory and Methods 1 (8)
ANOVA example
Rats on different doses of THC, k = 5 groups.
Overall sample mean y = 45.84
SSG =
k
X
ni (yi y )2
i=1
= 10 (34.00 45.84)2
+ 9 (51.56 45.84
MATH263: Statistical Theory and Methods 1 (4)
Two sample tests: Matched pairs
Often, want to compare two samples.
Special case: Two sets of observations on the same set of observational units (e.g.
MATH263: Statistical Theory and Methods 1 (3)
Hypothesis testing
Formulate a null hypothesis H0 and an alternative H1 , e.g.
H1 : 6= 0
H0 : = 0,
Collect the data and calculate an appropriate test st
MATH263: Statistical Theory and Methods 1 (2)
Statistical inference
Two kinds of question:
1. Estimation
Point estimation
Confidence interval
2. Hypothesis testing
Parametric (Normal distribution)
MATH263: Statistical Theory and Methods 1 (7)
ANOVA hypothesis test
Want to test whether group means i are all the same.
That is, see if differences between group sample means yi are big enough to b
MATH263: Statistical Theory and Methods 1 (5)
Comparison of two population variances
Two-sample t-test assumes both populations have the same (unknown) variance.
How can we check this?
Sample 1: X1
MATH263: Statistical Theory and Methods 1 (10)
Simple Linear Regression
Estimating the parameters 0 , 1 is useful, but not enough.
Also need confidence intervals for 0 , 1 , and need to be able to t
MATH263: Statistical Theory and Methods 1 (1)
Continuous random data
A random variable X can be
Discrete: The possible values of X are separated and individually distinct, e.g. X
cfw_0, 1, 2, . . .