## MATHS 2103 - Probability & Statistics II

### North Terrace Campus - Semester 1 - 2015

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 Probability & Statistics II School of Mathematical Sciences Semester 1 Undergraduate North Terrace Campus 3 Up to 3.5 hours per week Y MATHS 1012 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 David Green

##### 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.

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

University Graduate Attribute Course Learning Outcome(s)
Knowledge and understanding of the content and techniques of a chosen discipline at advanced levels that are internationally recognised. 1,2,3,4,5,6,7
The ability to locate, analyse, evaluate and synthesise information from a wide variety of sources in a planned and timely manner. 2
An ability to apply effective, creative and innovative solutions, both independently and cooperatively, to current and future problems. 2,6,7
Skills of a high order in interpersonal understanding, teamwork and communication. 2
A proficiency in the appropriate use of contemporary technologies. 1,2,3,4,5,6,7
A commitment to continuous learning and the capacity to maintain intellectual curiosity throughout life. 1,2,3,4,5,6,7
• Learning Resources
None.
##### 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 "Mathematical Statistics with Applications" by Wackerly, Mendenhall and Schaeffer (Duxbury 2008).
2. "Introduction to Stochastic Models" by Roe Goodman (2nd edition, Dover, 2006).
3. "Introduction to Probability Models" by Sheldon Ross (Academic Press, 2010).
4. "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
A semblance of the course notes will available online for those who wish to download and print prior to attending lectures. The format (either as two or one slide per page) is the same as the presentation slides used in the lectures, with room for you to annotate during lectures.

Recordings of lectures will also be available online immediately following each lecture for those who are unable to attend due to other commitments and for revision purposes.

All assignments, tutorials, handouts and solutions where appropriate will also be available online progressively as the course ensues.
• Learning & Teaching Activities
##### Learning & Teaching Modes
The lecturer will guide the students through the material presented in this course in a total of 36 lectures. Downloading and prereading the online notes will enable the students to more actively engage the material and interact during lectures. Students are expected to attend all lectures, but these will be recorded and made available online for those who are occasionally absent.

There are six tutorials during the presentation of the course that are held fortnightly, where groups of students will present solutions to assigned questions at the beginning of their tutorial. There will be sufficient time for active discussion on these questions where necessary and for the consideration of other questions.

In alternative weeks to the tutorial sessions, students will have assignment questions to submit for assessment which will be returned within two weeks, giving students direct feedback on their understanding of the course material. This material will contribute towards the students' final assessment as noted in the assessment summary.

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

 Activity Quantity Workload hours Lectures 36 90 Tutorials 6 18 Assignments 5 25 Group project 1 25 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.

The first tutorial in week 2 covers material from week 1 and other material that should be considered revision. Tutorials in weeks 4,6,8,10 and 12 cover the material of the previous two weeks.
##### Small Group Discovery Experience
There is a project, taken by groups of 3, which constitutes 10% of the final mark for this course.  This year it will be organised as a small group discovery experience component of the course.
• 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 70% All Assignments Formative and summative Weeks 3,5,7,9 and 11 20% All Group project Formative and summative Week 10 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 3% Assignment 2 Week 4 Week 5 3% Assignment 3 Week 6 Week 7 3% Assignment 4 Week 8 Week 9 3% Group Project Week 6 Week 10 15% Assignment 5 Week 10 Week 11 3%
##### Submission
Assignments must be submitted on time with a signed assessment cover sheet attached to the assignment. 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)
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 student’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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