37161 Probability and Random Variables
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Subject handbook information prior to 2024 is available in the Archives.
Credit points: 6 cp
Result type: Grade and marks
Requisite(s): (35101 Introduction to Linear Dynamical Systems OR 37131 Introduction to Linear Dynamical Systems) AND (35151 Introduction to Statistics OR 26134 Business Statistics OR 37151 Introduction to Statistics)) OR ((33130c Mathematics 1 OR 68103 Mathematics for Secondary Education Statistics OR 33190 Mathematical Modelling for Science)
The lower case 'c' after the subject code indicates that the subject is a corequisite. See definitions for details.
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 provides an effective way to quantify and model uncertainty. This subject introduces concepts in probability such as dependent and independent events as well as conditional probability. The idea of modelling random events with distributions is introduced, including probability calculation, expectation, variance, generating functions, and order statistics for independent events. The subject concludes by considering discrete Markov chains.
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 concerning the study of probability and random variables. |
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| 2. | apply the concepts of probability and random variables to simple problems, using computational skills where appropriate. |
| 3. | extract key information from a text and interpret this in correct mathematical terminology. |
| 4. | Use an appropriate combination of formulae and prose to explain a solution to a 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
This subject also contributes specifically to the development of the following Graduate Attributes to prepare students for professional practice:
Graduate Attribute 1 – Disciplinary knowledge
Activities in this subject develop practical skills to analyse problems which are inherently random or uncertain. Both the formal mathematical framework of probability theory and many of its common applications in diverse applied fields are explored.
Graduate Attribute 2 - Research. Inquiry and Critical Thinking
Many tasks in this subject can be attempted through different correct mathematical methods and students are encouraged to explore, consider and evaluate different approaches. This reflects research-inspired thinking and critical inquiry.
Graduate Attribute 4 – Reflection, Innovation, Creativity
Problems in the subject are motivated by real-world applications of probability theory. Numerical solutions to the modelling of these problems must then be considered in the context of the original motivating examples and their implications or findings justified.
Graduate Attribute 5 – Communication
Assessable labwork problems include short questions requiring clear explanations of mathematical concepts in everyday English language. These tasks are assessed both for the correctness of the mathematical and statistical ideas explained and also in terms of the clarity of the written communication.
Teaching and learning strategies
Weekly online: One 1.5 hr lecture, one 2 hr laboratory. Lectures include multiple tasks for students to participate in – small simulations, estimates etc and chance for peer-discussions – before getting results/feedback from the lecturer.
The laboratory/tutorial classes will incorporate a range of teaching and learning strategies including discussion of pre-class readings and student in-class groupwork. Classes each week should be supported by at least five hours per week of individual or group study, developing and practising skills by doing all pre-class preparatory work (posted on Canvas) and many textbook questions etc.
Throughout the laboratory-based classes, solutions will be given and allowances made for students to reflect on their answers and see additional clarification or support as required.
The fortnightly hand-in exercises allow for students to receive frequent low-stakes feedback on their progress. Additional problems, both pen-and-paper-based and computer-based, will be posted through Canvas to allow students to explore further according to individual needs. Solutions will be freely available, including real-time feedback on the computer-based questions.
Content (topics)
Topics include:
- Random experiments, deterministic versus stochastic systems; Interpretations of probability;
- Conditional probability; Mutual exclusivity; Venn diagrams; Simple set theory and de Morgan’s laws;
- Bayes Theorem; Hidden state models; Interpretation of signals;
- Discrete random variables; Probability mass functions; cumulative probability functions; Expectation and variance;
- Applications including fair pricing of financial options and elementary game theory;
- Bernoulli and Binomial distributions; convolutions;
- Exponential distribution and its relation to Poisson distribution; Conditional distributions and conditional expectation; Functions of random variables; Order statistics of i.i.d. variables;
- Generating functions; Sums of random variables;
- Discrete Markov Chains;
- Classification of Markov Chains, reducibility, periodicity, recurrent/transient/absorbing states.
Assessment
Assessment task 1: Problem solving workshops and written reports
| 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 |
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| Objective(s): | This assessment task addresses subject learning objective(s): 1, 2, 3 and 4 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: | 35% |
| Criteria: |
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Assessment task 2: In-class Test
| 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 |
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| Objective(s): | This assessment task addresses subject learning objective(s): , 1, 2 and 3 This assessment task contributes to the development of course intended learning outcome(s): 1.1, 2.1, 4.1 and 5.1 |
| Type: | Quiz/test |
| Groupwork: | Individual |
| Weight: | 15% |
| Length: | 90 minutes |
| Criteria: |
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Assessment task 3: Final examination
| Intent: | This assessment task contributes to the development of the following graduate attributes: 1. Disciplinary Knowledge 2. Research, inquiry and critical thinking 5. Communication |
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| Objective(s): | This assessment task addresses subject learning objective(s): 1, 2, 3 and 4 This assessment task contributes to the development of course intended learning outcome(s): 1.1, 2.1 and 5.1 |
| Type: | Examination |
| Groupwork: | Individual |
| Weight: | 50% |
| Criteria: |
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Minimum requirements
In order to pass this subject, a student must achieve a final result of 50% or more.
Recommended texts
Grimmett G. and Welsh D. Probability: An Introduction. 2nd Edition, Oxford, 2014
Tsokos, C. P. Probability for Engineering, Mathematics and Science. 1st edition. Cengage Learning, 2011.