This course establishes crucial concepts and asymptotic results that are widely relied upon in probability and statistics. These include convergence in distribution and the Central Limit Theorem for probability, and optimal properties of estimators, particularly maximum-likelihood estimators, in statistics. It also provides an opportunity for students to read further into one topic related to the course and write an essay to summarise what they have learnt.

20 lectures (2 each week)

5 tutorials (fortnightly throughout the semester)

STATS4047 Principles of Probability and Statistics

120-minute, end-of-course examination (100%)

The aims of this course are:

■ to introduce and discuss the concept of convergence in the theory of random variables;

■ to establish the laws of large numbers and the Central Limit Theorem;

■ to discuss optimal properties of point estimators;

■ to establish the large-sample properties of maximum-likelihood estimation;

■ to introduce students to the EM algorithm.

By the end of this course students will be able to:

■ describe and contrast convergence in probability, convergence in distribution, convergence in quadratic mean and almost sure convergence;

■ state, use and prove various probabilistic inequalities;

■ state, prove and use the Weak Law of Large Numbers, the Strong Law of Large Numbers and the Central Limit Theorem;

■ state and discuss optimal properties of point estimators;

■ state, prove and use the Rao-Blackwell Theorem and the Cramer-Rao lower bound;

■ state, prove and apply general asymptotic properties of maximum-likelihood estimators;

■ construct an EM algorithm for various missing data problems.

■ prove Hoeffding's inequality

■ write down the expression for the best linear unbiased estimator (BLUE) and explain how it is derived..

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.