MATHS 2103 - Probability & Statistics II

North Terrace Campus - Semester 1 - 2021

Probability theory is the branch of mathematics that deals with modelling uncertainty. It is important because of its direct application in areas such as genetics, finance and telecommunications. It also forms the fundamental basis for many other areas in the mathematical sciences including statistics, modern optimisation methods and risk modelling. This course provides an introduction to probability theory, random variables and Markov processes. Topics covered are: probability axioms, conditional probability; Bayes' theorem; discrete random variables, moments, bounding probabilities, probability generating functions, standard discrete distributions; continuous random variables, uniform, normal, Cauchy, exponential, gamma and chi-square distributions, transformations, the Poisson process; bivariate distributions, marginal and conditional distributions, independence, covariance and correlation, linear combinations of two random variables, bivariate normal distribution; sequences of independent random variables, the weak law of large numbers, the central limit theorem; definition and properties of a Markov chain and probability transition matrices; methods for solving equilibrium equations, absorbing Markov chains.

  • General Course Information
    Course Details
    Course Code MATHS 2103
    Course Probability & Statistics II
    Coordinating Unit School of Mathematical Sciences
    Term Semester 1
    Level Undergraduate
    Location/s North Terrace Campus
    Units 3
    Contact Up to 3.5 hours per week
    Available for Study Abroad and Exchange Y
    Prerequisites MATHS 1012 or MATHS 1004
    Course Description Probability theory is the branch of mathematics that deals with modelling uncertainty. It is important because of its direct application in areas such as genetics, finance and telecommunications. It also forms the fundamental basis for many other areas in the mathematical sciences including statistics, modern optimisation methods and risk modelling. This course provides an introduction to probability theory, random variables and Markov processes.

    Topics covered are: probability axioms, conditional probability; Bayes' theorem; discrete random variables, moments, bounding probabilities, probability generating functions, standard discrete distributions; continuous random variables, uniform, normal, Cauchy, exponential, gamma and chi-square distributions, transformations, the Poisson process; bivariate distributions, marginal and conditional distributions, independence, covariance and correlation, linear combinations of two random variables, bivariate normal distribution; sequences of independent random variables, the weak law of large numbers, the central limit theorem; definition and properties of a Markov chain and probability transition matrices; methods for solving equilibrium equations, absorbing Markov chains.
    Course Staff

    Course Coordinator: Dr Andrew Black

    Course Timetable

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

  • Learning Outcomes
    Course Learning Outcomes
    Students who successfully complete this course should be able to demonstrate understanding of :
    1 basic probability axioms and rules and the moments of discrete and continous random variables as well as be familiar with common named discrete and continous random variables.
    2 how to derive the probability density function of transformations of random variables and use these techniques to generate data from various distributions.
    3 how to calculate probabilities, and derive the marginal and conditional distributions of bivariate random variables.
    4 discrete time Markov chains and methods of finding the equilibrium probability distributions.
    5 how to calculate probabilities of absorption and expected hitting times for discrete time Markov chains with absorbing states.
    6 how to translate real-world problems into probability models.
    7 how to read and annotate an outline of a proof and be able to write a logical proof of a statement.


    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)
    Deep discipline knowledge
    • informed and infused by cutting edge research, scaffolded throughout their program of studies
    • acquired from personal interaction with research active educators, from year 1
    • accredited or validated against national or international standards (for relevant programs)
    1,2,3,4,5,6,7
    Critical thinking and problem solving
    • steeped in research methods and rigor
    • based on empirical evidence and the scientific approach to knowledge development
    • demonstrated through appropriate and relevant assessment
    5,6,7
    Teamwork and communication skills
    • developed from, with, and via the SGDE
    • honed through assessment and practice throughout the program of studies
    • encouraged and valued in all aspects of learning
    4,5
    Career and leadership readiness
    • technology savvy
    • professional and, where relevant, fully accredited
    • forward thinking and well informed
    • tested and validated by work based experiences
    6
    Self-awareness and emotional intelligence
    • a capacity for self-reflection and a willingness to engage in self-appraisal
    • open to objective and constructive feedback from supervisors and peers
    • able to negotiate difficult social situations, defuse conflict and engage positively in purposeful debate
    4,5
  • Learning Resources
    Required Resources
    Access to the internet.
    Recommended Resources
    There are many good books on probability and statistics in the Barr Smith Library, with the following texts being recommended for this course.

    1. Probability and Random Processes, by Grimmett and Stirzaker, 3rd edition, Oxford, 2001
    2. Mathematical Statistics with Applications, by Wackerly, Mendenhall and Schaeffer, Duxbury, 2008.
    3. Introduction to Stochastic Models, by Roe Goodman, 2nd edition, Dover, 2006.
    4. Introduction to Probability Models, by Sheldon Ross, Academic Press, 2010.
    5. Mathematical Statistics and Data Analysis, by John Rice, Duxbury Press, 2006.

    For other texts on probability and statistics, try browsing books with call numbers beginning with 519.2.
    Online Learning
    This course uses MyUni exclusively for providing electronic resources, such as topic videos, lecture notes, assignment papers, and sample solutions. Students should make appropriate use of these resources.
  • 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
    Topics Schedule
    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.



    Each tutorial, starting in week 2, will cover material from the previous week.
    Small Group Discovery Experience
    NA
  • 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 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


    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.

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  • Policies & Guidelines
  • Fraud Awareness

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