Undergraduate Programme and Module Handbook 2023-2024

# Module MATH2647: PROBABILITY II

## Department: Mathematical Sciences

### MATH2647: PROBABILITY II

Type | Open | Level | 2 | Credits | 10 | Availability | Available in 2023/24 | Module Cap | Location | Durham |
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#### Prerequisites

- (Calculus I (Maths Hons) (MATH1081) or Calculus I (MATH1061) and Probability I (MATH1597)) and Linear Algebra I (Maths Hons) (MATH1091) or Linear Algebra I (MATH1071) and Analysis I (MATH1051) [the latter may be a co-requisite].

#### Corequisites

- Analysis I (MATH1051) unless taken before.

#### 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.

#### Content

- Probability spaces and probability measures.
- Infinite collections of events.
- Random variables and expectation.
- Infinite sequences of random variables.
- Modes of convergence.
- Limit theorems
- Generating functions.
- Further topics to be chosen from: random graphs, branching processes, probability and algorithms.

#### Learning Outcomes

Subject-specific Knowledge:

- 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 and a workig knowledge of generating functions and their computational and theoretical power.
- Reproduce theoretical mathematics concerning sequences of events and sequences of random variables to a level appropriate to Level 2, including key definitions and theorems..

Subject-specific Skills:

- In addition students will have emhanced mathematical skills in the following areas: intuition for probabilistic reasoning.

Key Skills:

- 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

- Lecturing demonstrates 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, revealling also the thought processes behind such solutions.
- Tutorials provide active problem-solving engagement and immediate feedback to the learning processes.
- Formative assessments provide feedback to guide students in the correct development of their knowledge and skills in preparation for the summative assessment.
- The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

#### Teaching Methods and Learning Hours

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

Lectures | 21 | 2 per week in Epiphany; 1 in Easter | 1 Hour | 21 | |

Tutorials | 5 | Fortnightly in Epiphany; 1 in Easter | 1 Hour | 5 | ■ |

Problems Classes | 4 | Fortnightly in Epiphany | 1 Hour | 4 | |

Preparation and Reading | 70 | ||||

Total | 100 |

#### Summative Assessment

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

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

End of year written examination | 2 hours | 100% | Yes |

#### Formative Assessment:

Weekly written or electronic assignments to be assessed and returned.

■ 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