MATHS 2103 - Probability & Statistics II
North Terrace Campus - Semester 1 - 2023
General Course Information
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 Coordinator: Dr Adam - Benjamin Rohrlach
The full timetable of all activities for this course can be accessed from Course Planner.
Course Learning OutcomesStudents 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)
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.
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.
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.
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.
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.
Required ResourcesAccess to the internet.
Recommended ResourcesThere 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 LearningThis 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 ModesThis 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.
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.
The University's policy on Assessment for Coursework Programs is based on the following four principles:
- Assessment must encourage and reinforce learning.
- Assessment must enable robust and fair judgements about student performance.
- Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
- Assessment must maintain academic standards.
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 RequirementsAn aggregate score of 50% is required in order to pass this course.
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%
SubmissionAssignments 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.
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.
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