MA4PAM: Probability and Measure

Module code: MA4PAM

Module provider: Mathematics and Statistics; School of Mathematical, Physical and Computational Sciences

Credits: 20

Level: Level 4 (Undergraduate Masters)

When you'll be taught: Semester 1

Module convenor: Dr Jochen Broecker, email: j.broecker@reading.ac.uk

Pre-requisite module(s): BEFORE TAKING THIS MODULE YOU MUST ( TAKE MA2RA2 OR TAKE MA2RAT ) AND TAKE MA1FM (Compulsory)

Co-requisite module(s):

Pre-requisite or Co-requisite module(s):

Module(s) excluded: IN TAKING THIS MODULE YOU CANNOT TAKE MA3PAM OR TAKE MA3MTI (Compulsory)

Placement information: NA

Academic year: 2024/5

Available to visiting students: Yes

Talis reading list: Yes

Last updated: 21 May 2024

Overview

Module aims and purpose

To introduce students to measure theory, integration, and measure theoretic probability. Measure theory and integration develops a notion of integration that is more powerful than Riemann integration and behaves much better in connection with limits of functions. The theory is therefore fundamental to further topics in analysis. Furthermore, measure theory and integration forms the rigorous basis of a probability theory in which the number of possible events is not restricted to be finite. This will form the second part of the course. 

Module learning outcomes

By the end of the module, it is expected that students will be able to:

1. Demonstrate a systematic understanding of knowledge, and a critical awareness of current problems and/or new insights regarding the integration of measurable functions, including interchange of integral and limits of functions
2. Demonstrate self-direction and originality, and act autonomously in tackling and solving problems related to conditional probabilities and conditional expectation
3. Deal with complex issues both systematically and creatively in the use of limit theorems such as the law of large numbers, the ergodic theorem, the central limit theorem, and martingale limit theorems 
4. Learn independently as is required for continuous personal development in the area of measure theory, measure theoretic probability, and their applications throughout other branches of mathematics

Module content

Part 1: Measure theory and integration

• Definitions and basic properties of sigma algebras and measurable functions
• Definitions and basic properties of measures
• Sequences and limits of sets and their probabilities
• Caratheodory’s theorem and construction of the Lebesgue measure
• The integral
• Monotone convergence, bounded (dominated) convergence, Fatou’s lemma
• Lp spaces and their completeness
• Transformations and push—forward
• Products spaces and product measures, Fubini-Theorem

Part 2: Probability theory

• Translation into probability-theoretic language
• Distributions and independence
• Conditional probabilities and conditional expectations
• Martingales and martingale limit theorem
• Characteristic functions and the Central Limit Theorem
• Stationary processes and the ergodic theorem

Structure

Teaching and learning methods

The material is delivered via lectures supported by tutorials with formative exercises. 

Study hours

At least 55 hours of scheduled teaching and learning activities will be delivered in person, with the remaining hours for scheduled and self-scheduled teaching and learning activities delivered either in person or online. You will receive further details about how these hours will be delivered before the start of the module.

Please note the independent study hours above are notional numbers of hours; each student will approach studying in different ways. We would advise you to reflect on your learning and the number of hours you are allocating to these tasks.

Semester 1 The hours in this column may include hours during the Christmas holiday period.

Semester 2 The hours in this column may include hours during the Easter holiday period.

Summer The hours in this column will take place during the summer holidays and may be at the start and/or end of the module.

Assessment

Requirements for a pass

Students need to achieve an overall module mark of 50% to pass this module.

Summative assessment

Penalties for late submission of summative assessment

The Support Centres will apply the following penalties for work submitted late:

Assessments with numerical marks

• where the piece of work is submitted after the original deadline (or any formally agreed extension to the deadline): 10% of the total marks available for that piece of work will be deducted from the mark for each working day (or part thereof) following the deadline up to a total of three working days;
• the mark awarded due to the imposition of the penalty shall not fall below the threshold pass mark, namely 40% in the case of modules at Levels 4-6 (i.e. undergraduate modules for Parts 1-3) and 50% in the case of Level 7 modules offered as part of an Integrated Masters or taught postgraduate degree programme;
• where the piece of work is awarded a mark below the threshold pass mark prior to any penalty being imposed, and is submitted up to three working days after the original deadline (or any formally agreed extension to the deadline), no penalty shall be imposed;
• where the piece of work is submitted more than three working days after the original deadline (or any formally agreed extension to the deadline): a mark of zero will be recorded.

Assessments marked Pass/Fail

• where the piece of work is submitted within three working days of the deadline (or any formally agreed extension of the deadline): no penalty will be applied;
• where the piece of work is submitted more than three working days after the original deadline (or any formally agreed extension of the deadline): a grade of Fail will be awarded.

The University policy statement on penalties for late submission can be found at: /cqsd/-/media/project/functions/cqsd/documents/qap/penaltiesforlatesubmission.pdf

You are strongly advised to ensure that coursework is submitted by the relevant deadline. You should note that it is advisable to submit work in an unfinished state rather than to fail to submit any work.

Formative assessment

Formative assessment is any task or activity which creates feedback (or feedforward) for you about your learning, but which does not contribute towards your overall module mark.

Non-assessed problem sheets 

Reassessment

Additional costs