Module MATH3211: PROBABILITY III

Department: MATHEMATICAL SCIENCES

MATH3211: PROBABILITY III

Prerequisites

• Complex Analysis II (MATH2011) and Analysis in Many Variables II (MATH2031); alternatively, Contours and Symmetries II (MATH2111) or Contours and Hyperbolic Geometry II (MATH2**1) AND Analysis in Many Variables II (MATH2031).

Corequisites

• None.

Excluded Combination of Modules

• Probability IV (MATH4131).

Aims

• To build a logical structure on probabilistic intuition, and to cover such peaks of the subject as the Strong Law of Large Numbers and the Central Limit Theorem, as well as more modern topics such as Martingale Theory.

Content

• Probability spaces revisited: Infinite sample spaces, random variables.
• Probability distribution functions: joint distributions, characteristic functions, application to sums of independent Normal, Poisson, gamma RV's.
• Convergence of random variables: Monotone and dominated convergence, Borel-Cantelli lemmas, the strong law of large numbers.
• Central limit theorem.
• Martingale theory: conditional expectation, Radon-Nikodym Theorem, upcrossings, martingale convergence theorem, optional stopping, Wald's lemma, fair games, applications.

Learning Outcomes

• By the end of the module students will: be able to solve novel and/or complex problems in Probability.
• have a systematic and coherent understanding of theoretical mathematics in the field of Probability.
• have acquired coherent body of knowledge of these subjects demonstrated through one or more of the following topic areas: Probability as a measure.
• Random variables.
• Convergence Theorems.
• Probability under partial information.
• Applications of Probability.

• In addition students will have specialised mathematical skills in the following areas which can be used with minimal guidance: 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.
• Assignments for self-study develop problem-solving skills and enable students to test and develop their knowledge and understanding.
• Formatively assessed assignments provide practice in the application of logic and high level of rigour as well as feedback for the students and the lecturer on students' progress.
• 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:

Four written assignments to be assessed and returned. Other assignments are set for self-study and complete solutions are made available to students.

■ 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