Unit of study_

# STAT4028: Probability and Mathematical Statistics

Probability Theory lays the theoretical foundations that underpin the models we use when analysing phenomena that involve chance. This unit introduces the students to modern probability theory and applies it to problems in mathematical statistics. You will be introduced to the fundamental concept of a measure as a generalisation of the notion of length and Lebesgue integration which is a generalisation of the Riemann integral. This theory provides a powerful unifying structure that bring together both the theory of discrete random variables and the theory of continuous random variables that were introduce to earlier in your studies. You will see how measure theory is used to put other important probabilistic ideas into a rigorous mathematical framework. These include various notions of convergence of random variables, 0-1 laws, and the characteristic function. You will then synthesise all these concepts to establish the Central Limit Theorem and also verify important results in Mathematical Statistics. These involve exponential families, efficient estimation, large-sample testing and Bayesian methods. Finally you will verify important convergence properties of the expectation-maximisation (EM) algorithm. By doing this unit you will become familiar with many of the theoretical building blocks that are required for any in-depth study in probability or mathematical statistics.

Code STAT4028 Mathematics and Statistics Academic Operations 6
 Prerequisites: ? None None STAT4528 STAT3X23 or equivalent: that is, a sound working and theoretical knowledge of statistical inference

At the completion of this unit, you should be able to:

• LO1. Demonstrate a coherent and advanced knowledge of the concepts of measure theory and Lebesgue integration and how they provide a unified approach to a wide variety of problems arising in probability.
• LO2. Communicate mathematical analyses and solutions to mathematical and practical problems in probability and related fields clearly in a variety of media to diverse audiences.
• LO3. Utilise characteristic function techniques to prove foundational theoretical results in probability.
• LO4. Compare and contrast different forms of stochastic convergence.
• LO5. Construct optimal solutions to a wide variety of problems in mathematical statistics.
• LO6. Devise solutions to novel mathematical problems in probability theory.

## Unit outlines

Unit outlines will be available 2 weeks before the first day of teaching for the relevant session.