Course Outline
Course Outline
Course Outline
ACTL2131
Probability and Mathematical Statistics
Course Outline
Semester 1, 2016
business.unsw.edu.au
CRICOS Code 00098G
Dear Students,
Welcome to ACTL2131 Probability and Mathematical Statistics. This course is
one of eight courses covering the Core Technical subjects of the Institute of
Actuaries offered in BActSt. Many of you will also be completing ACTL2111
Financial Mathematics for Actuaries. In the early weeks of the courses you will
find that you will have to adjust to the study load in the courses and also take
some time to note the links between these courses.
Probability is the branch of science concerned with the mathematical models
and techniques for making quantitative inferences about uncertainty. Statistics
deals with the collection and the analysis of data. Together, these two
disciplines provide us with fundamental tools to analyse risk and manage the
financial consequences of future uncertainties. This course provides you with a
foundation in the probability models and statistical techniques required for
analysing risks in modern financial markets. It provides a foundation for courses
in your actuarial major in the final years of undergraduate study at UNSW. I
hope you find the course challenging and interesting.
The way the course is taught this year is radically different. It is often
referred to as flipped.
The main rationale for this new structure is to bring the face-to-face time later in
the learning process, when students are more comfortable with the materials,
and more likely to interact and ask questions. The first conceptual encounter
with the materials happens at home when students watch video lectures. They
then move on to practicing their knowledge with tutorial exercises. At this stage,
tutorial sessions provide some face-to-face on a weekly basis, to personalised
help. Consultation is also available. Towards the end of the learning of a given
module, everyone gathers in the lecture room for a lectorial. The word
combines lecturesbecause they are run by the lecturer, and with the whole
group, and tutorialbecause their goal is not to lecture students, but to
discuss a module at a higher conceptual level, and to cement students learning
with other activities (such as guest lectures, discussions, advanced exercises).
Please read it carefully and thoroughly, as it will be assumed that you are
familiar with the contents.
If you have any questions about the course at any time then please contact me.
Katja Ignatieva (LIC, course coordinator)
business.unsw.edu.au
CRICOS Code 00098G
Table of Contents
PART A: COURSE-SPECIFIC INFORMATION
COURSE DETAILS
3
4
4
4
5
6
ASSESSMENT
7
8
9
10
10
10
12
COURSE RESOURCES
12
13
COURSE SCHEDULE
14
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Room
Telephone
Consultation times
Dr. Katja
Ignatieva (Course
coordinator,
lecturer in
charge)
k.ignatieva@unsw.e
du.au
Business
School, Room
651
9385 6810
Friday, 11-12pm,
RitchieTh
Katja is responsible for the administration and final assessment of the course, as well
as the lectures and related teaching and learning.
Tutors for ACTL 2131 are:
Staff
Nikolay Gudkov
n.gudkov@unsw.edu.au
Vincent Tu
v.tu@unsw.edu.au
Hayden Lau
kawai.lau@unsw.edu.au
They are responsible for the tutorials and grading of mid-semester test and assignment
assessment tasks.
Who should I contact?
1. Questions about the video lectures and lectorials: Katja Ignatieva during
the lectorials and consultation times;
2. Questions about tutorial problems: the tutors during the tutorials and
consultation times (will be announced prior to mid-term and final exams);
3. Administrative enquiries about the course: Katja Ignatieva, during the
consultation times or by e-mail;
4. Enquiries about coursework programs in Actuarial Studies: School office
(rasadmin@unsw.edu.au);
5. Enrolment: Business School Student Centre:
http://www.asb.unsw.edu.au/currentstudents/resources/businessstudentce
ntre/Pages/default.aspx
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2 COURSE DETAILS
2.1
ColomboThA
RitchieTh
Stream B:
Friday
Note: Consultation for both streams takes place weekly on Friday, 11:00 a.m. 12:00
p.m in RitchieTh.
Timetables and locations are correct at time of printing. A full timetable of lectures and
topics is provided later in this course study guide. Any alterations to the lecture times or
locations will be advised in lectures and via Moodle.
Students should consult Moodle on a regular basis, since assignment questions and
other course materials will be placed there. The web address is:
http://moodle.telt.unsw.edu.au/
Tutorials
Students must attend the tutorial for which they are enrolled. Attendance will be
recorded and count towards meeting the requirements to pass the course. If you wish
to change your tutorial then you must lodge an application to change your tutorial time
with the Business School student service centre.
In tutorials, we will implement interactive learning where collaborative group work is
highly encouraged. To get the most out of the tutorials, students should read lecture
notes and textbooks and references and complete assigned homework problems in
advance of the tutorial.
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Peer-Assisted-Support-Scheme (PASS):
In addition, ASOC has an actuarial PASS program which is available to enhance
student learning in this course. It is highly recommended that you attend these
sessions.
ASOC website: http://www.asoc.unsw.edu.au/
PASS peer support class times:
http://www.asoc.unsw.edu.au/index.php?option=com_content&view=article&id=48&Ite
mid=42
PASS material is available on https://sites.google.com/a/asoc.unsw.edu.au/unswasoc/downloads
2.2
Units of Credit
2.3
Summary of Course
This course covers probability and statistics topics relevant to actuarial studies. Topics
covered include univariate and multivariate random variables, moments, moment
generating functions, marginal and conditional distributions, sampling distributions,
estimation methods, hypothesis tests, and linear regression. Examples relevant to
actuarial studies, finance and insurance are used to illustrate the application of the
topics covered.
2.4
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Students should have a solid background in mathematics and are assumed to be able
to use a computer to analyse financial problems. You should be able to use a word
processing package (such as WORD) and a spreadsheet (such as EXCEL).
2.5
The aims of Section 2.3 (A to D) have been broken down into the following learning
outcomes. At the end of the course students should be able to:
A1. Explain the concepts of probability.
A2. Explain the concepts of random variable, probability distribution, distribution
function, expected value, variance and higher moments, and calculate expected
values and probabilities associated with the distributions of random variables.
A3. Define a moment generating function, derive it in simple cases, and use it to
evaluate moments
A4. Define basic discrete and continuous distributions, be able to apply them and
simulate them in simple cases.
A5. Explain the concepts of independence, jointly distributed random variables and
conditional distributions, and use generating functions to establish the
distribution of linear combinations of independent random variables.
A6. Explain the concepts of conditional expectation and compound distribution, and
apply them.
A7. State the central limit theorem, and apply it.
B1. Summarise the main features of a data set (exploratory data analysis).
B2. Explain the concepts of random sampling, statistical inference and sampling
distribution, and state and use basic sampling distributions.
C1. Describe the main methods of estimation and the main properties of estimators,
and apply them.
C2. Construct confidence intervals for unknown parameters.
D1. Test hypothesis.
E1. Investigate linear relationships between variables using correlation analysis and
regression analysis.
The following table shows how your Course Learning Outcomes relate to the overall
Program Learning Goals and Outcomes, and indicates where these are assessed (they
may also be developed in tutorials and other activities):
Program Learning Goals
and Outcomes
Course Assessment
Item
Exercise
Mid-semester
exam
Assignment
Exam
Exercise
Mid-semester
Knowledge
of
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modelling.
exam
Assignment
Exam
3a
Written
communication
Assignment
3b
Oral communication
Active lectures
participation
Teamwork
Assignment
Active lectures
participation
5a.
Not specifically
course.
addressed
in
this
5b.
Not specifically
course.
addressed
in
this
2.6
This course corresponds largely with the actuarial professional subject CT3 Probability
and Mathematical Statistics. The courses Learning Outcomes relate to the aims of this
Institute of Actuaries course in the following way:
Course Learning Outcomes
A1
CT3: ii,
A2
CT3: iii
A3
CT3: iv
A4
CT3: v
A5
CT3: vi
A6
CT3: xiv
A7
CT3: vii
B1
CT3: i
B2
CT3: viii
B3
CT3: ix
B4
CT3: x
B5
CT3: xi
B6
CT3: xii
B7
CT3: xiii
C1
None
D1
None
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The approach adopted in this course is one of assisted self-study. The approach
adopted in this course is called flipped classroom. While reading this subsection,
please refer to the schedule given in Section 7.
The main rationale for this new structure is to bring the face-to-face time later in
the learning process, when students are more comfortable with the materials,
and more likely to interact and ask questions. The first conceptual encounter
with the materials happens at home when students watch the video lectures.
They then move on to practicing their knowledge with tutorial exercises. At this
stage, tutorial sessions provide some face-to-face, and personalised help.
Consultation is also available. Towards the end of the learning of a given
module, everyone gathers in the lecture room for a lectorial. The word
combines lecturesbecause they are run by the lecturer, and with the whole
group, and tutorialbecause their goal is not to lecture students, but to
discuss a module at a higher conceptual level, and to cement students learning
with other activities (such as guest lectures, discussions, advanced exercises).
Course materials are organised in 4 modules. Students are responsible to learn topics
with the following materials:
Prescribed textbooks (and recommended books for additional support)
Video lectures available on the course website
Tutorial exercises with solutions
Past quizzes and exams for advanced exercises
Additionally, students who are not familiar with the software R should complete the
module R you ready? (with videos, exercises and documents) which is available on
the course website to all ACTL students.
The philosophy underpinning this course and its Teaching and Learning Strategies are
based on Guidelines on Learning that Inform Teaching at UNSW. These guidelines
may be viewed at: www.guidelinesonlearning.unsw.edu.au. Specifically, the lectures,
tutorials and assessment have been designed to appropriately challenge students and
support the achievement of the desired learning outcomes. A climate of inquiry and
dialogue is encouraged between students and teachers and among students (in and
out of class). The lecturers and tutors aim to provide meaningful and timely feedback to
students to improve learning outcome. This is not a course where you can become
proficient just by observing.
You will need to get involved in class - evaluating information, asking and answering
questions. You also must learn to organize your independent study and practice
enough problems to gain a thorough understanding of concepts and how to apply
them.
This course is very condensed and the subjects each week require that you know the
material of past weeks. Therefore, falling behind will lead to less effective lectures and
tutorials and is thus not recommended.
The course puts emphasis on statistics, i.e., parameter estimation, hypothesis testing
and linear regression (week 5-12). In order to understand statistics one should have
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some knowledge about probability (week 1-4). The probability part of the course is
taught in only four weeks, which means that the workload in those weeks might be
larger than in the second part of the course. To ease the workload, you are expected to
make exercises during the tutorials. The number of examples in the lecture notes has
increased substantially. In the lecture you are required to make attempts to these
examples, some of them are to do at home.
3.2
It is expected that the students will take a pro-active approach to learning. The course
is organised in the following learning activities.
Self-study
During the time periods of self-study, students should cover the readings, video
lectures and tutorials for the associated module. A required learning strategy for this
course is to have read all prescribed readings, watched the associated video lectures
and attempted the tutorial exercises before lectures.
Lectorials
The purpose of lectorials is to provide a logical structure for the topics that make up the
course, to emphasize the important concepts and methods of each topic, and to
provide relevant examples to which the concepts and methods are applied. Lectorials
will wrap up modules and provide opportunities to ask questions about the associated
modules.
Tutorials
The more you read the more you know, but the more you practice the more you
learn and understand. So the key to the understanding of this course is problem
solving.
The purpose of tutorials is to enable you to raise questions about difficult topics or
problems encountered in their studies. Students must not expect another lecture, but
must attempt the questions themselves in groups and the tutor will answer questions to
the group, not to the whole class.
A required learning strategy for the tutorials (on which provision of the course materials
is based) is:
Prior to make an attempt of the exercises, review your lecture notes and videos.
Prior to the tutorial, make an attempt to the exercises you should make before the
tutorial (see Section 7: Course Schedule).
During the tutorial, make an attempt to the exercises you should make in the
tutorial (see Section 7: Course Schedule).
After the tutorial, make an attempt to the exercises you should after in the tutorial
(see Section 7: Course Schedule).
If you have questions about the tutorial exercises, ask them from your tutor. If you
think you have a good understanding of the material, you should try and answer the
questions of your peers. This will give you feedback on your ability to explain the
material and hence how well you know the material.
Check your answer using the tutorial solution.
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The teaching strategy (including the feedback loop) in this course is:
Get introduced
Try it out
Get feedback
Try again
The Get introduced (explanation of course concepts in the lecture) and Try it out
(examples in the lecture) are part of the lecture. If you are not able to make satisfactory
attempt to the examples in the lecture, this is feedback that you should revise the
lecture material in depth after the lecture. The Try again (tutorial exercises) and Get
feedback (answers from tutor/ tutorial solutions) are part of the tutorial.
The tutorial is designed to attempt the tutorial questions in groups (typically of 3-4
students). This collaborative working is advised since it allows learning from peers and
allows the more advanced student with a possibility to test whether s/he knows the
material in depth and is thereby able to explain it to peers. If the group is unable to
solve a question, it can ask for help from the tutor. The tutor will not provide the
answer, but would help you in the direction of the solution. This is because you should
practice and learn by doing, rather than seeing the solution. At the end of the week the
tutorial solutions will be posted on Moodle. It will only be posted at the end of the week
to give you time to attempt the questions without a solution manual. The solution
manual should not be part of attempting a question, but to verify whether your attempt
was correct.
A required learning strategy for the tutorials (on which provision of the course materials
is based) is:
Prior to make an attempt of the exercises, review your lecture notes.
Prior to the tutorial, make an attempt to the exercises you should make before the
tutorial (see Section 7: Course Schedule).
During the tutorial, make an attempt to the exercises you should make in the
tutorial (see Section 7: Course Schedule).
After the tutorial, make an attempt to the exercises you should after in the tutorial
(see Section 7: Course Schedule).
If you have questions about the tutorial exercises, ask them from your tutor. If you
think you have a good understanding of the material, you should try and answer the
questions of your peers. This will give you feedback on your ability to explain the
material and hence how well you know the material.
Check your answer using the tutorial solution.
3.3
The course ACTL 2131 Probability and Mathematical Statistics is a condense course.
To optimize the learning experience we will use several learning and teaching
technologies.
Moodle
This course will use Moodle for communication with students.
Moodle will contain the course outline, lecture notes, homework and tutorial exercises,
assessment information, and any notices relevant to this course. It is important that you
visit the site regularly to see any notices posted there by the course coordinator. The
site can be accessed at: http://moodle.telt.unsw.edu.au/
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Video Lectures
You are expected to view the videos on Moodle before attending the lecture, see
course schedule. The material covered in the videos will not be covered in the lectures,
except when there are clarification questions from students. The lecture will provide
tutorial style examples (i.e., using collaborative group work in lecture) of the material
covered in the video lectures. Therefore, it is essential that you study this material
before the lecture.
Turnitin
Turnitin will be used for handing in the assignment. More information will be given in
the documentation for the assignment.
Some resources for help:
Turnitin tutorials:
http://www.turnitin.com/en_us/training/student-training
Turnitin help centre:
http://www.turnitin.com/en_us/support/help-center/general-articles
4 ASSESSMENT
4.1
Formal Requirements
4.2
Assessment Details
Assessment of your performance in the course will be done through a number of tasks.
These are listed in the table below.
Assessment
Task
Weight
LO1
Mid-term
15%
Weeks 1-4:
A1-A6, B1
1 hour
Group Assignment
20%
A1-A7, B1,
C1, D1
NA
Final examination
65%
All
2 hours
Length
Due Date
Thursday, 7th April
16.00-18.00.
Location: ClancyTh
Theatre
Friday 27 May
11am sharp
TBA
Mid-term
Technical skills are important in practice and this course provides foundation technical
skills that will be useful throughout your working life.
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10
In order to assess your understanding of the technical skills covered in the course there
will be a 1-hour quiz during the session. The quiz will be closed book. Students will only
be allowed to bring the textbook "Formulae and Tables for Actuarial Examinations" into
the quiz, which should be fully unannotated!
Normal examination rules apply to the conduct of the quiz. Calculators will be allowed
in the quiz and the final examination but a clear indication of all of the steps involved in
your calculations must be shown. The University will not supply calculators to students
for use in examinations where the provision of calculators has not been requested by
the course examiner. It is the students responsibility to be familiar with the rules
governing the conduct of examinations.
The mid-term requires written responses, with students earning marks for correct
mathematical working as well as part marks for incorrect responses with correct
method and reasoning. They test not only your knowledge of the material, but also the
depth of your understanding of it.
Group Assignment
There will be one group assignment for this course. One group member is to submit the
assignment on behalf of the team. The assignment offers students the opportunity to
engage in independent research, engage in critical analysis, self-reflection and
problem-solving, as well as to demonstrate their understanding of the concepts and
perspectives that are central to actuarial studies. Students are reminded that the work
they submit must be their own (see course outline part B). While we have no problem
with students working together on the assignment problems, the material students
submit for assessment must be their own.
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11
Avoid a 0 for your assignment (in the mildest case) because of plagiarism
Students are reminded that the work they submit must be their own (see section 5
above). While we have no problem with students working together on the assignment
problems, the material students submit for assessment must be their own. This means
that:
1. The mathematical solutions you present are written up by you, without
reference to any other students work.
2. Students should make sure they understand what plagiarism is (see Section 5
and do the quiz) cases of plagiarism have a very high probability of being
discovered. For issues of collective work, having different persons marking the
assignment does not decrease this probability.
Students should consult the Turnitin section of the website accessible to all ACTL
students well in advance, as this gives a (non exhaustive) list of things that could go
wrong and explains how the policies above are implemented.
Final Examination
The final examination will assess students understanding of the concepts covered in
the course and their ability to apply them to probability and statistics problems.
The final examination will be a two-hour written paper. The final examination will be
closed book. Students will only be allowed to bring the text "Formulae and Tables for
Actuarial Examinations" into the exam. This must not be annotated. Students may bring
their own calculators. All calculators must be either UNSW or Actuarial Studies
approved.
4.3
Late Submission
The School of Risk and Actuarial Studies has a policy of grading late assignments with
a zero mark. Punctual submission of work is required in order to satisfy the
requirements of the course. The assignment may be marked at the discretion of the
course co-ordinator if there is a valid reason for late submission and used in cases
where your final overall results are marginal.
Quality Assurance
The Business School is actively monitoring student learning and quality of the
student experience in all its programs. A random selection of completed
assessment tasks may be used for quality assurance, such as to determine the
extent to which program learning goals are being achieved. The information is
required for accreditation purposes, and aggregated findings will be used to
inform changes aimed at improving the quality of Business School programs. All
material used for such processes will be treated as confidential.
5 COURSE RESOURCES
Textbooks
The prescribed textbooks for the course are:
1. [FT] The Faculty of Actuaries and The Institute of Actuaries (2002), Formulae
and tables for examinations of the Faculty of Actuaries and The Institute of
Actuaries. (The formulae book you can use, if unannotated, in quizzes and
exams for actuarial courses.)
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12
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13
7 COURSE SCHEDULE
Note: Tutorials start in Week 1 and finish in Week 12.
This timetable below may change. Revisions will be advised as they occur through the course
web site.
COURSE SCHEDULE
Week
Self-study
Lectorial
Tutorial
Week 1: 29 February
Module 1
Introduction
Module 1
Week 2: 7 March
Module 1
Module 1
Module 1
Week 3: 14 March
Module 1
Week 4: 21 March
Module 1
Module 1
Module 1
Module 1
Module 2
Module 2
Module 2
Week 7: 18 April
Module 3
Week 8: 25 April
Module 3
Week 9: 2 May
Module 3
Module 4
Module 4
Module 2
Module 2
Module 3
Module 3
Module
Module 3
Module 3-4
Module 3-4
Module 4
Module 4
Module 4
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14
Modules
Topics
Prliminaries Revision
Probability space
Calculating with probablity
Counting
1 Probability Theory
1.1
Mathematical methods
Random variables
Measures of location
Measure of dispersion
Moments (central/non-central)
Generating functions
1.2
Univariate Ditributions
Bernoulli Distribution
Binomial Distribution
Geometric Distribution
Negative Binomial Distribution
Poisson Distribution
Exponential Distribution
Gamma Distribution
Normal Distribution
Uniform Distribution
Beta Distribution
Weibull Distribution
Pareto Distribution
[W+]
[JR]
2.1-2.10
Chapters 2, 3, 4
2.11, 3.1-3.2, 4.1-4.2, 4.11
3.3, 4.3
3.3, 4.3
3.9, 4.9
3.9, 3.10
Chapters 3, 4
3.4
3.4
3.5
3.6
3.8
4.6
4.6
4.5
4.4
4.7
IAAust CT3
Week
ii.1-3
ii.1-7
ii.3
1
1
1
iii.1-2
i.2, iii.3, vi.6
i.3, iii.3, vi.6
i.4, iii.3
iv.1-5, vi.7
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
15
Modules
Topics
1.3
Joint and Multivariate Distributions
Bivariate distribution functions
Mean, variance, covariance & correlation
Conditional Distributions
Bivariate Normal Distribution
Law of iterated expectation
Conditional Variance Identity
Multivarate distribution functions
1.4
1.5
[W+]
Chapter 5
5.1-5.4
5.5-5.8
5.11
5.1
5.11
5.11
5.9
2.12, 7.1
6.7
1.1-1.5
7.1-7.2
1.1-1.5
Chapter 6
6.1-6.3
6.4, 6.6
6.5
5.1-5.4
7.1-7.2
[JR]
IAAust CT3
Week
vi.1-3, vi.5
vi.4, vi.6
vi.1
vi.1-2
xiv.1-2
xiv.1-2
vi.1-2
3
3
3
3
3
3
3
i.2, viii.1-3
4
4
4
4
4
v.4
v.4
4
4
4
4
4
vi.4
v.2, viii.5-7
16
Modules
Topics
2 Parameter Estimation
2.1
Estimation techniques
Introducition/Definitions
Method of Moments
Maxium Likelihood Estimation (MLE)
Bayesian Estimation
2.2
2.3
[W+]
8.1-8.4
9.6
9.7, 9.8
[JR]
IAAust CT3
Week
xi.3-5
xi.1
xi.2
5
5
5
5
Limit theorems
Chebyshev's inequality
Convergence concepts
Law or Large Numbers
Central Limit Theorem
7.3
7.3-7.4
7.3-7.4
vii.1
vii.2
vii.2-3
5
5
5
5
Evaluating Estimators
UMVUE's
Cramer-Rao Low Bound
Consistent and sufficient statistics
Confidence Intrevals
Properties of MLEs
9.1-9.2
9.5
9.3-9.4
8.5-8.9
8.7-8.9
xi.3-5
xi.3-5
xi.3-5
x.1-6
xi.6
6
6
6
6
6
17
Modules
Topics
3 Hypothesis Testing
3.1
Statistical test procedure
Selection of the null hypothesis, alternative hypothesis
Regection region
Best critical region
Neuman Pearson Lemma
Uniformly most powerful test
Generalized Likelihood Ratio test
3.2
3.3
3.4
3.5
[W+]
[JR]
IAAust CT3
10.1-10.2
10.3, 10.5
10.3, 10.7
10.10
10.10
10.11
10.2, 10.4
10.10
10.6, 10.8, 10.9
Parametric tests
Fisher's exact test
Contingency table
Chi-2 goodness of fit test
r-sample multinomial test
14.5
14.4-14.7
14.1-14.3
14.4-14.7
Nonparametric tests
One-sample sign test
Two-sample sign test
15.1-15.3
15.1-15.3
Week
7
7
7
7
7
7
xi.1-3
xi.1-3
xi.1-3
xi.3
8
8
8
8
8
9
9
9
9
xi.4
xi.4
xi.4
9
9
Goodness-of-fit test
Anderson-Darling &Kolmogorov-Smirnov test
9
9
18
Modules
Topics
4 Linear Regression
4.1
Simple Linear Regression
Correlation coefficient
Assumptions
Relationship between MLE and LSE
Partitioning the variability
4.2
4.3
4.4
[W+]
[JR]
IAAust CT3
Week
11.8
11.1-11.2
11.3-11.4
13.1-13.2
xi.1-4
xi.1-4
xi.5
xi.7, xi.9
10
10
10
10
11.5, 11.9
11.5, 11.9
11.6, 11.9
11.7, 11.9
11.8, 11.9
xi.6
xi.6
xi.8
xi.8
xi.7
10
10
10
10
10
11.1
11.11
11.12
11.12
11.13-11.15
xi.9
xi.9
xi.9
xi.6
xi.8
11
11
11
11
11
12
12
12
12
19