Undergraduate Programme and Module Handbook 2018-2019 (archived)

# Module MATH4131: PROBABILITY IV

## Department: Mathematical Sciences

### MATH4131: PROBABILITY IV

Type | Open | Level | 4 | Credits | 20 | Availability | Available in 2018/19 | Module Cap | Location | Durham |
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#### Prerequisites

- Complex Analysis II (MATH2011) AND Analysis in Many Variables II (MATH2031) AND Probability II (MATH2647) AND 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.

#### Content

- Introductory examples.
- Coin tossing and trajectories of random walks.
- Discrete renewal theory.
- Limit theorems and convergence.
- Order statistics.
- Non-classical limits and their applications.
- Stochastic order.
- Additional topics in advanced probability.
- (NB the syllabus is identical to PROBABILITY III (A) which is taught in parallel).

#### Learning Outcomes

Subject-specific Knowledge:

- 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:
- Random walks.
- Convergence theorems.
- Discrete renewal theory.
- Advanced applications of probability.

Subject-specific Skills:

- In addition students will have highly specialised and advanced mathematical skills in the following areas: Modelling, Computation.

Key Skills:

- Students will be able to study independently to further their knowledge of an advanced topic.

#### 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.
- Subject material assigned for independent study develops the ability to acquire knowledge and understanding without dependence on lectures.
- 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. The Subject material assigned for independent study will form part of the examined material.

#### Teaching Methods and Learning Hours

Activity | Number | Frequency | Duration | Total/Hours | |
---|---|---|---|---|---|

Lectures | 42 | 2 per week for 20 weeks and 2 in term 3 | 1 Hour | 42 | |

Problems Classes | 8 | Four in each of terms 1 and 2 | 1 Hour | 8 | |

Preparation and Reading | 150 | ||||

Total | 200 |

#### Summative Assessment

Component: Examination | Component Weighting: 100% | ||
---|---|---|---|

Element | Length / duration | Element Weighting | Resit Opportunity |

Written examination | 3 Hours | 100% |

#### Formative Assessment:

Eight 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