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.

20 lectures (2 each week)

5 tutorials (fortnightly throughout the semester)

STATS5022 Principles of Probability and Statistics (Level M)

90-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 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.

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