Applied Probability III

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Description

Many processes in the real world involve some random variation superimposed on a deterministic structure. For example, the experiment of ¿ipping a coin is best studied by treating the outcome as a random one. Mathematical probability has its origins in games of chance with dice and cards, originating in the ¿fteenth and sixteenth centuries. This course aims to provide a basic toolkit for modelling and analyzing discrete-time problems in which there is a significant probabilistic component. We will consider Markov chain examples in the course including population branching processes (with application to genetics), random walks (with application to games), and more general discrete time examples using Martingales.

Objective

to introduce students to the basic tools used by applied probabilists, and to give students a thorough grounding in the theory of discrete-time Markov chains on a countable state space. To provide a thorough grounding in discrete-time Markov chains on a countable state space. The student will obtain an in-depth knowledge of Markov chains both from a theoretical and a modelling viewpoint.

Content

Topics covered are: basic probability and measure theory, discrete time Markov chains, hitting probabilities and hitting time theorems, population branching processes, inhomogeneous random walks on the line, solidarity properties and communicating classes, necessary and sufficient conditions for transience and positive recurrence, global balance, partial balance, reversibility, Martingales, stopping times and stopping theorems with a link to Brownian motion.

David Green Lecturer for this course

Delivery

36 hours of lectures and tutorials.

Assessment

Ongoing assessment 30%, exam 70%.

Graduate attributes

• Applying mathematical knowledge (1)
• Interpreting data and drawing conclusions (2)
• Formulating mathematical problems (3)
• Problem solving skills (4)
• Flexibility in working environment (5)
• Team participation and leadership (7)

Linkage past

PREREQUISITE: MATHS 1012 Mathematics IB. ASSUMED KNOWLEDGE: Markov Chains as taught in MATHS 2103 Probability & Statistics.

Linkage present

A sister unit is APP MTH 3016 Telecommunications Systems Modelling III. Other units relating to Operations Research are APP MTH 3014 Optimisation III, APP MTH 3005 Mathematical Programming !III and APP MTH 3010 Variational Methods and Optimal Control III.

Linkage future

The material covered in this unit is useful for APP MTH 3016 Telecommunications Systems Modelling III.

Restrictions

None.

Recommended text

References: a list will be provided in lectures