Undergraduate Programme and Module Handbook 2015-2016 (archived)
Module MATH4131: PROBABILITY IV
Department: Mathematical Sciences
MATH4131:
PROBABILITY IV
Type |
Open |
Level |
4 |
Credits |
20 |
Availability |
Available in 2016/17 and alternate years thereafter |
Module Cap |
|
Location |
Durham
|
Prerequisites
- Complex Analysis II (MATH2011) AND (Probability and
Actuarial Mathematics II (MATH2161) OR Probability and Geometric
Topology II (MATH2151) ) AND Analysis in Many Variables II (MATH2031)
AND a minimum of 40 credits of Mathematics modules at Level
3.
Corequisites
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
Activity |
Number |
Frequency |
Duration |
Total/Hours |
|
Lectures |
40 |
2 per week for 19 weeks and 2 in term 3. |
1 Hour |
40 |
|
Problems Classes |
8 |
Four in each of terms 1 and 2 |
1 Hour |
8 |
|
Preparation and Reading |
|
|
|
152 |
|
Total |
|
|
|
200 |
|
Summative Assessment
Component: Examination |
Component Weighting: 100% |
Element |
Length / duration |
Element Weighting |
Resit Opportunity |
three-hour examination |
|
100% |
|
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