37262 Mathematical Statistics
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Subject handbook information prior to 2025 is available in the Archives.
Credit points: 6 cp
Result type: Grade and marks
Requisite(s): (37161 Probability and Random Variables AND (33230 Mathematics 2 OR 33290 Statistics and Mathematics for Science OR 37132 Introduction to Mathematical Analysis and Modelling)) OR (37161 Probability and Random Variables AND 26134 Responsible Evidence-Based Decisions)
These requisites may not apply to students in certain courses. See access conditions.
Anti-requisite(s): 35363 Stochastic Models
Description
When modelling real-world problems we need to deal with uncertainty, and probability and statistics provide an effective way to quantify and explain uncertainty. This subject covers more advanced topics in both probability and frequentist statistics, including goodness-of-fit tests and regression models, and explores the mathematical and conceptual foundations of these as well as their application. Finally, Bayesian statistics is introduced alongside modern procedures commonly employed for Bayesian statistics, such as Markov Chain Monte Carlo (MCMC) methods.
Subject learning objectives (SLOs)
Upon successful completion of this subject students should be able to:
| 1. | Understand, explain and prove the principal ideas and results which underpin the fields of probability and statistics. |
|---|---|
| 2. | Apply skills in theoretical and computational techniques of simulation modelling to solve problems tractable with these techniques |
| 3. | Formulate a mathematical model for problems expressed in everyday language |
| 4. | Interpret the output of a mathematical simulation and express this in context-appropriate language |
| 5. | Apply critical and creative thinking to find a valid approach to address complex problems. |
| 6. | Assess the suitability of a model for a given problem |
Course intended learning outcomes (CILOs)
This subject also contributes specifically to the development of following course intended learning outcomes:
- Demonstrate theoretical and technical knowledge of mathematical sciences including calculus, discrete mathematics, linear algebra, probability, statistics and quantitative management. (1.1)
- Evaluate mathematical and statistical approaches to problem solving, analysis, application, and critical thinking to make mathematical arguments, and conduct experiments based on analytical, numerical, statistical, algorithms to solve new problems. (2.1)
- Design creative solutions to contemporary mathematical sciences-related issues by incorporating innovative methods, reflective practices and self-directed learning. (4.1)
- Use succinct and accurate presentation of reasoning and conclusions to communicate mathematical solutions, and their implications, to a variety of audiences, using a variety of approaches. (5.1)
Contribution to the development of graduate attributes
Contribution to the development of graduate attributes
This subject provides students with knowledge and skills to prepare them for professional practice. This subject contributes to the development of the following Science graduate attributes:
1. Disciplinary knowledge
Activities in this subject develop both practical and technical skills in the area of mathematical simulation. The laboratory/practical classes focus on the implementation of these concepts and the three in-class tests assess students’ understanding of exactly how and when these techniques are appropriate and applicable.
2. Research, Inquiry and critical thinking
Throughout the subject, in both the lectures/seminars and the laboratory classes, students are encouraged to consider problems which are formulated in unfamiliar terms. This allows students the opportunity to explore and to discuss a number of different approaches and evaluate the merits of each. In many cases, students may realise that there are multiple equally-valid approaches to tackling a given problem. The laboratory-practical classes provide clear illustrations of the wide-ranging applications and power of simulation-based approaches to problem solving. Students work on case studies from fields as diverse as finance, genetics, engineering and conservation biology
4. Reflection, Innovation, Creativity
For many of the industry-inspired problems in the subject, multiple mathematical approaches are possible. Students are encouraged to make reasoned judgements on the merits of each and, where appropriate, explore new options and mathematical techniques.
5. Communication
The practical/laboratory classes foster skills in communicating to diverse audiences (expert and non-expert) through scenarios presented in everyday (non-mathematical) language which can be formulated, modelled and simulated. The results of these simulations must be appropriately presented in clear language and context appropriate to the initial problem.
Teaching and learning strategies
Lectures/seminars will be posted as videos (youtube and downloadable mp4/PowerPoint files) and can be viewed at any time. These will be accompanied by weekly live Zoom sessions.
Lab classes will operate on campus (2hr per week) with provisions made for students who, owing to COVID restrictions are unable to attend in person.
Subject delivery (per week):
- One 2hr seminar/lecture
- One 2hr practical/laboratory class
- Pre-class preparatory readings via Canvas
The lecture/seminars will be largely interactive, presenting student with the opportunity to engage with higher-order thinking activities and receive feedback from the lecturer. In additional to presentation of the underlying theoretical work, frequent problems will be demonstrated.
To derive maximum benefit from the classes, students should preview the preparatory material, complete practical exercises and review the materials presented in seminars/lectures. Students are also encouraged to seek independently additional resources and background information from both the UTS Library and Canvas resources.
The material assessed in the quizzes will be all technical material covered in the seminars/lectures. Students should spend a minimum of two hours each week preparing the underlying mathematics of simulation as well as practising its implementation.
Throughout the laboratory-based classes, students will have the opportunity to interact with other students and with the laboratory tutor to receive frequent guidance, feedback and opportunity for reflection. Each week, feedback and assessment marks from the previous week will be available to each student.
Content (topics)
The major topics in this subject include:
- Simulation of random variables
- Multivariate distributions
- Distributions of functions of random variables
- t, chi-square and F distributions
- Markov and Chebyshev inequalities. Weak law of large numbers
- Estimation. Method of moments and maximum likelihood estimation
- Linear regression
- Confidence and prediction intervals
- Generalised linear models. Exponential family
- Markov Chain Monte Carlo (MCMC). Metropolis-Hastings algorithm
- Bayesian inference and conjugate priors
Assessment
Assessment task 1: Quizzes
| Intent: | This assessment task contributes to the development of the following graduate attributes: 1. Disciplinary Knowledge 2. Research, inquiry and critical thinking |
|---|---|
| Objective(s): | This assessment task addresses subject learning objective(s): 2, 3, 4 and 6 This assessment task contributes to the development of course intended learning outcome(s): 1.1 and 2.1 |
| Type: | Quiz/test |
| Groupwork: | Individual |
| Weight: | 40% |
| Length: | 75 minutes |
| Criteria: | 1. Correct choice of reasoning 2. Correct application of knowledge and procedures 3. Correct choice of problem solving strategies and procedures 4. Clear communication using correct mathematical terminology, including assessment of the plausibility of the results obtained. |
Assessment task 2: Weekly labwork
| Intent: | This assessment task contributes to the development of the following graduate attributes: 1. Disciplinary Knowledge 2. Research, inquiry and critical thinking 4. Reflection, Innovation, Creativity 5. Communication |
|---|---|
| Objective(s): | This assessment task addresses subject learning objective(s): 1, 2, 3, 4, 5 and 6 This assessment task contributes to the development of course intended learning outcome(s): 1.1, 2.1, 4.1 and 5.1 |
| Type: | Laboratory/practical |
| Groupwork: | Individual |
| Weight: | 60% |
| Length: | 30 minutes (preceded by 90 minutes of interactive labwork with peer and tutoror support.) |
| Criteria: |
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Minimum requirements
In order to pass this subject, a student must achieve a final result of 50% or more.