Module MATH2751: Probability II

Department: Mathematical Sciences

MATH2751: Probability II

Prerequisites

• One of:
• Calculus I (Maths Hons) (MATH1081) OR Calculus I (MATH1061)
• AND one of:
• Linear Algebra I (Maths Hons) (MATH1091) OR Linear Algebra I (MATH1071)
• AND:
• Probability I (MATH1597)
• AND:
• Analysis I (MATH1051) [may be taken as a co-requisite]

Corequisites

• Analysis 1 (MATH1051) if not taken at Level 1.

Excluded Combination of Modules

• None

Aims

• To reinforce the knowledge of probability gained at Level 1, develop probabilistic ideas and techniques in more sophisticated settings, and to provide a firm foundation for modules in this area in higher years.
• To introduce and develop the concept of a Markov chain, as a fundamental type of stochastic process, and to study key features of Markov models using probabilistic tools such as generating functions.

Content

• Probability spaces. events, probability measures, and random variables.
• Countable collections of events and the Borel-Cantelli lemma.
• Infinite sequences of random variables, modes of convergence, and laws of large num-bers
• Introduction to Lebesgue approach to expectation; monotone and dominated convergence.
• Generating functions.
• Markov chains, the Markov property, and classification of states.
• Hitting probabilities, stopping times, and the strong Markov property.
• Qualitative long-time behaviour: recurrence and transience.
• Convergence to equilibrium and stationary distributions.
• Further topics to be chosen from: random walks, Gibbs sampler, coupling, mixing and card shuffling, stochastic epidemics, discrete renewal theory, Markov decision processes.

Learning Outcomes

• By the end of the module students will:
• Be able to solve seen and unseen problems on the given topics.
• Have a knowledge and understanding of this subject demonstrated through an ability to establish probabilistic results for sequences of events and sequences of random variables, to identify stationary distributions, classify states, and compute hitting probabilities and expected hitting times, and a working knowledge of generating functions and their computational and theoretical power.
• Be able to reproduce theoretical mathematics concerning sequences of events, sequences of random variables, and Markov chains, including key definitions and theorems.

• In addition students will have enhanced mathematical skills in the following areas: intuition for probabilistic reasoning and key features of probabilistic systems that evolve in time.

• Students will have basic mathematical skills in the following areas: problem solving, modelling, computation.

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

• Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
• Problem classes show how to solve example problems in an ideal way, revealing also the thought processes behind such solutions.
• Tutorials provide active problem-solving engagement and immediate feedback to the learning process.
• Formative assessments provide feedback to guide students in the correct development of their knowledge and skills in preparation for the summative assessment.
• Summative assignments test achievement of learning outcomes and provide feedback to students about their mastery of the topics.
• The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

Teaching Methods and Learning Hours

Summative Assessment

Formative Assessment:

Fortnightly assignments.

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University