Module MATH4131: PROBABILITY IV

Department: Mathematical Sciences

MATH4131: PROBABILITY IV

Prerequisites

• (Complex Analysis II (MATH2011) OR Contours and Symmetries II (MATH2111) OR Contours and Hyperbolic Geometry II (MATH2121) OR Contours and Actuarial Mathematics II (MATH2171) OR Contours and Probability II (MATH2**1)) AND (Probability and Actuarial Mathematics II (MATH2161) OR Probability and Geometric Topology II (MATH2151) OR Contours and Probability II (MATH2**1) OR Codes and Probability II (MATH2**1)) AND Analysis in Many Variables II (MATH2031). In addition, a minimum of 40 credits of Mathematics modules at Level 3.

Corequisites

• None.

Excluded Combination of Modules

• Probability III (MATH3211).

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 distributions: 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, upcrossings, martingale convergence theorem, optional stopping, Wald's lemma, fair games, applications.
• (NB the syllabus is identical to PROBABILITY III (A) which is taught in parallel).

Learning Outcomes

• By the end of the module students will: be able to solve complex, unpredictable and specialised problems in Probability.
• have an understanding of specialised and complex theoretical mathematics in the field of Probability.
• have mastered a 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 highly specialised and advanced mathematical skills in the following areas: 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 complex and specialised 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