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Fundamentals of Probability Theory

Key information

  • Module code:

    6CCM341A

  • Level:

    6

  • Semester:

      Autumn

  • Credit value:

    15

Module description

Syllabus:

Countability, measure spaces, σ-algebras, π-systems and uniqueness of extension. Construction of Lebesgue measure on R (proof non-examinable), Independence.

The Borel-Cantelli lemmas, measurable functions and random variables, independence of random variables. Notions of probabilistic convergence. Construction of integral and expectation. Integration and limits. Density functions. Product measure and Fubini’s theorem. Laws of large numbers.

Characteristic functions and weak convergence, Gaussian random variables. The central limit theorem. Conditional probability and expectation.

Prerequisites: 

4CCM141A/5CCM141B Probability and Statistics I, 5CCM241A/6CCM241B Probability and Statistics II, 5CCM221A Real Analysis (advisable)

Assessment details

Written examination.

Educational aims & objectives

Aims:

This course provides a rigorous introduction to the mathematics underlying probability theory. We will make sense of the notion of probability and expectation through the concept of measure theory (which has many other applications in analysis, e.g. partial differential equations). If you enjoy building a substantial and coherent mathematical theory, you will enjoy the measure theory part of this course. Probability will provide motivation and applications throughout the course and we will prove some key results in probability and statistics, such as the Strong Law of Large Numbers and Central Limit Theorem.

Unlike Probability and Statistics I & II, this course has a heavy analysis flavour with many proofs.

Learning outcomes

On successful completion of this module students will

  • understand the concepts of sigma fields and probability measures on general probability spaces
  • be able to characterise and use random variables with general distributions
  • be able to use the Lebesgue measure and Lebesgue-Stieltjes integration
  • understand and work with different modes of convergence in probability
  • State and use laws of large numbers and the central limit theorem
  • understand conditional probability and expectation

Teaching pattern

Three hours of lectures and one hour of tutorial per week throughout the term

Department

Module description disclaimer

King’s College London reviews the modules offered on a regular basis to provide up-to-date, innovative and relevant programmes of study. Therefore, modules offered may change. We suggest you keep an eye on the course finder on our website for updates.

Please note that modules with a practical component will be capped due to educational requirements, which may mean that we cannot guarantee a place to all students who elect to study this module.

Please note that the module descriptions above are related to the current academic year and are subject to change.