## MATHS 7103 - Probability & Statistics PG

### North Terrace Campus - Semester 1 - 2023

In this course, you will get an introduction to probability theory, random variables and Markov processes. You will learn how to deal with modelling uncertainty, which has direct real-world application in areas such as genetics, finance and telecommunications. It is also the basis for many other areas in mathematical sciences, including statistics, modern optimisation methods and risk modelling.

• General Course Information
##### Course Details
Course Code MATHS 7103 Probability & Statistics PG Mathematical Sciences Semester 1 Postgraduate Coursework North Terrace Campus 3 Up to 3.5 hours per week Y MATHS 7027 Ongoing assessment, exam
##### Course Staff

Course Coordinator: Dr Adam - Benjamin Rohrlach

##### Course Timetable

The full timetable of all activities for this course can be accessed from Course Planner.

• Learning Outcomes
##### Course Learning Outcomes
On successful completion of this course you will be able to:
 1 Demonstrate an understanding of basic probability axioms; the rules and moments of discrete and continuous random variables. 2 Derive the probability density function (PDF) of transformations of random variables and generate data from various distributions. 3 Calculate probabilities and derive the marginal and conditional distributions of bivariate random variables. 4 Find equilibrium probablity distributions. 5 Calculate probabilities of absorption and expected hitting times for discrete time Markov chains with absorbing states.

##### University Graduate Attributes

This course will provide students with an opportunity to develop the Graduate Attribute(s) specified below:

University Graduate Attribute Course Learning Outcome(s)

Attribute 1: Deep discipline knowledge and intellectual breadth

Graduates have comprehensive knowledge and understanding of their subject area, the ability to engage with different traditions of thought, and the ability to apply their knowledge in practice including in multi-disciplinary or multi-professional contexts.

1,2,3,4,5,6,7

Attribute 2: Creative and critical thinking, and problem solving

Graduates are effective problems-solvers, able to apply critical, creative and evidence-based thinking to conceive innovative responses to future challenges.

5,6,7

Attribute 3: Teamwork and communication skills

Graduates convey ideas and information effectively to a range of audiences for a variety of purposes and contribute in a positive and collaborative manner to achieving common goals.

4,5

Attribute 4: Professionalism and leadership readiness

Graduates engage in professional behaviour and have the potential to be entrepreneurial and take leadership roles in their chosen occupations or careers and communities.

6

Attribute 8: Self-awareness and emotional intelligence

Graduates are self-aware and reflective; they are flexible and resilient and have the capacity to accept and give constructive feedback; they act with integrity and take responsibility for their actions.

4,5
• Learning & Teaching Activities
##### Learning & Teaching Modes
This course relies on topic videos as the primary delivery mechanism for the material. Tutorials are the primary direct contact hours, during which students will both reinforce and employ the understanding obtained through lectures. Weekly quizzes provide regular opportunities for students to gauge their progress and understanding.
##### Workload

The information below is provided as a guide to assist students in engaging appropriately with the course requirements.

 Activity Quantity Workload hours Videos and study 88 Tutorials 11 33 Assignments 5 25 Online quizzes 12 12 Total 158
##### Learning Activities Summary
 Week 1 Discrete random variables Probability mass function, expectation and variance. Bernoulli distribution, Geometric distribution, Binomial distribution. Derivation of mean and variance. Week 2 Discrete random variables Sampling with and without replacement. Hypergeometric distribution and Poisson distribution. Derivation of the Poisson distribution as limiting form of Binomial. Derivation of mean and variance. Week 3 Discrete random variables Bounding probabilities, tail sum formula, Markov’s inequality and Chebyshev’s inequality. Probability generating functions and moment generating functions. Week 4 Continuous random variables Probability density function, cumulative distribution function, expectation, mean and variance. Moment generating functions and uniqueness theorem. Chebyshev’s inequality. Week 5 Continuous random variables The uniform distribution on (a, b), the normal distribution. Mean and variance of the normal distribution. The Cauchy distribution. The exponential distribution, moments, memoryless property, hazard function. Week 6 Continuous random variables Gamma distribution, moments, Chi-square distribution. Point processes, the Poisson process, derivation of the Poisson and exponential distributions. Week 7 Transformation of random variables and bivariate distributions Cumulative distribution function method for finding the distribution of a function of random variable. The transformation rule. Discrete bivariate distributions, marginal and conditional distributions, the trinomial distribution and multinomial distribution. Week 8 Bivariate distributions Continuous bivariate distributions, marginal and conditional distributions, independence of random variables. Covariance and correlation. Mean and variance of linear combination of two random variables. The joint Moment generating function (MGF) and MGF of the sum. Week 9 Bivariate distributions and independent random variables The bivariate normal distribution, marginal and conditional distributions, conditional expectation and variance, joint MGF and marginal MGF. Linear combinations of independent random variables. Means and variances. Sequences of independent random variables and the weak law of large numbers. The central limit theorem, normal approximation to the binomial distribution. Week 10 Discrete time Markov chains Definition of a Markov chain and probability transition matrices. Equilibrium behaviour of Markov chains: computer demonstration and ergodic, limiting and stationary interpretations. Week 11 Discrete time Markov chains Methods for solving Equilibrium Equations using probability generating functions and partial balance. Week 12 Discrete time Markov chains Definition of absorbing Markov chains, structural results, hitting probabilities and expected hitting times. Review.

Tutorials, starting in week 2, will cover material from the previous week(s).
• Assessment

The University's policy on Assessment for Coursework Programs is based on the following four principles:

1. Assessment must encourage and reinforce learning.
2. Assessment must enable robust and fair judgements about student performance.
3. Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
4. Assessment must maintain academic standards.

##### Assessment Summary
 Assessment task Task type When due Weighting Learning outcomes Examination (3 hours) Summative Examination period 50% All Assignments Formative and summative Weeks 3,5,7,9 and 11 20% All Mid-semester test Formative and summative Week 7 20% All Online quizzes Formative and summative Weeks 1-12 10% All

Note that the examination for MATHS 7103 is of 3 hours duration.

##### Assessment Related Requirements
An aggregate score of 50% is required in order to pass this course.
##### Assessment Detail
 Assessment task Set Due Weighting Assignment 1 Week 2 Week 3 4% Assignment 2 Week 4 Week 5 4% Assignment 3 Week 6 Week 7 4% Assignment 4 Week 8 Week 9 4% Assignment 5 Week 10 Week 11 4%
##### Submission
Assignments are submited electronically through MyUni. Late assignments will not be accepted. Assignments will be returned within two weeks. Students may be excused from an assignment for medical or compassionate reasons. In such cases, documentation is required and the lecturer must be notified as soon as possible before the fact.
##### Course Grading

Grades for your performance in this course will be awarded in accordance with the following scheme:

M10 (Coursework Mark Scheme)
Grade Mark Description
FNS   Fail No Submission
F 1-49 Fail
P 50-64 Pass
C 65-74 Credit
D 75-84 Distinction
HD 85-100 High Distinction
CN   Continuing
NFE   No Formal Examination
RP   Result Pending

Further details of the grades/results can be obtained from Examinations.

Grade Descriptors are available which provide a general guide to the standard of work that is expected at each grade level. More information at Assessment for Coursework Programs.

Final results for this course will be made available through Access Adelaide.

• Student Feedback

The University places a high priority on approaches to learning and teaching that enhance the student experience. Feedback is sought from students in a variety of ways including on-going engagement with staff, the use of online discussion boards and the use of Student Experience of Learning and Teaching (SELT) surveys as well as GOS surveys and Program reviews.

SELTs are an important source of information to inform individual teaching practice, decisions about teaching duties, and course and program curriculum design. They enable the University to assess how effectively its learning environments and teaching practices facilitate student engagement and learning outcomes. Under the current SELT Policy (http://www.adelaide.edu.au/policies/101/) course SELTs are mandated and must be conducted at the conclusion of each term/semester/trimester for every course offering. Feedback on issues raised through course SELT surveys is made available to enrolled students through various resources (e.g. MyUni). In addition aggregated course SELT data is available.

• Student Support
• Policies & Guidelines
• Fraud Awareness

Students are reminded that in order to maintain the academic integrity of all programs and courses, the university has a zero-tolerance approach to students offering money or significant value goods or services to any staff member who is involved in their teaching or assessment. Students offering lecturers or tutors or professional staff anything more than a small token of appreciation is totally unacceptable, in any circumstances. Staff members are obliged to report all such incidents to their supervisor/manager, who will refer them for action under the university's studentâ€™s disciplinary procedures.

The University of Adelaide is committed to regular reviews of the courses and programs it offers to students. The University of Adelaide therefore reserves the right to discontinue or vary programs and courses without notice. Please read the important information contained in the disclaimer.

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