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Probability 34 (MATH M0700)

Contents of this document:

• Administrative information
• Unit aims,   General Description, and Relation to Other Units
• Teaching methods and Learning objectives
• Assessment methods and Award of credit points
• Transferable skills
• Texts and Syllabus

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Administrative Information

1. Unit number and title: MATH M0700 Probability 34
2. Level: M/7
3. Credit point value: 10 credit points
4. Year: 12/13
5. First Given in this form: 2008-9
6. Unit Organiser: Erwan Hillion
7. Lecturer: Dr Erwan Hillion
8. Teaching block: 2
9. Prerequisites: MATH11300 Probability 1 and MATH21400 Applied Probability 2

Unit aims

To outline, discuss, and prove some of the key results in probability theory and their applications to statistics.

General Description of the Unit

This course deals with various modes of convergence of random variables (almost surely, weak, in probability, and in L^p) and the connections between them. We also discuss and prove weak laws and strong laws of large numbers, prove Borel-Cantelli lemmas, Kolmogorov 0-1 law, and the three series theorem. We study the properties and applications of characteristic function. Central Limit Theorems, Lindeberg conditions, and time permitting Local limit theorems and Barry-Essen inequality.

Relation to Other Units

This unit develops the rigorous theoretical background to much of probabilistic (and partly statistical) methodology covered in probability/statistics units at levels 1, 2, 3, and M.

Teaching Methods

Lectures, assignments, and exercises to be done by students.

Learning Objectives

To gain a (better) understanding of:

• Conditional expectations;
• Types of convergence of random variables;
• Strong and Weak laws and Central Limit Theorems;
• The ways to establish the above results rigorously.

Assessment Methods

20% of the assessment mark for Probability 34 is based on the assignment done during the course.

80% of the assessment mark is calculated from a 1½-hour written examination in April consisting of THREE questions. A candidate's best TWO answers will be used for assessment. Calculators are NOT permitted for this examination.

 

Award of Credit Points

Examination mark of 50 or more.

Transferable Skills

Self-assessment by working examples sheets and using solutions provided.

Texts

Each of the following texts will be useful:

1. R. Durrett, Probability: Theory and Examples, 2nd edition, Duxbury Press.
2. S.R.S. Varadhan, Probability Theory, AMS
3. G.R. Grimmett and D.R. Stirzaker, Probability and Random Processes, Oxford Univ. Press.

Syllabus

1. Properties of conditional expectations.
2. Modes of convergence of random variables.
3. Borel-Cantelli lemmas, 0-1 laws.
4. Weak and Strong law of large numbers.
5. Characteristic functions.
6. Central Limit Theorems and related topics.