LEARNING OUTCOMES
This course equips you with a deep and rigorous understanding of probability theory. It provides the mathematical tools and abstract concepts necessary to model and analyse complex probabilistic phenomena, apply advanced statistical techniques, and engage in research and applications in mathematics, statistics, machine learning, and other fields. What you will learn:
- Develop a strong understanding of probability spaces and sigma-algebras, enabling you to operate with random numbers, vectors, and sequences as measurable functions.
- Compute probabilities, expected values, variances, and correlations by integrating with respect to a probability measure.
- Apply products of probability measures and probability kernels to model and analyse stochastic independence and conditional probabilities.
- Get introduced to key convergence concepts of random sequences (almost sure, in probability) and classical limit laws of random sequences.
- Become familiar with key statistical distances (total variation, Wasserstein) and information-theoretic divergences (eg Kullback–Leibler) to quantify the difference between two probability distributions.
Credits: 5
Schedule: 02.09.2024 - 14.10.2024
Teacher in charge (valid for whole curriculum period):
Teacher in charge (applies in this implementation): Lasse Leskelä
Contact information for the course (applies in this implementation):
CEFR level (valid for whole curriculum period):
Language of instruction and studies (applies in this implementation):
Teaching language: English. Languages of study attainment: English
CONTENT, ASSESSMENT AND WORKLOAD
Content
valid for whole curriculum period:
This course provides a unified foundation of probability theory in a general measure-theoretic framework where random variables are represented as functions on an abstract measurable space that is equipped with a probability measure from which all probabilities, expected values, and correlations associated with a given stochastic model can be computed. Key concepts:
- Random numbers, vectors, and sequences as measurable functions.
- Integration with respect to a general probability measure.
- Probability kernels for modelling conditional probabilities.
- Convergence of random sequences and probability measures.
- Distances and divergences between probability measures.
Assessment Methods and Criteria
valid for whole curriculum period:
Exam and homeworks.
Workload
valid for whole curriculum period:
Participation in contact teaching (2 x 2h weekly lectures, 1 x 2h weekly exercise classes) and independent study.
DETAILS
Study Material
valid for whole curriculum period:
The main study material consists of lecture notes and homework problems delivered during the course. Most of the material is also available in textbooks:
- J Jacod & P Protter. Probability Essentials. Springer 2004.
- D Williams. Probability with Martingales. Cambridge University Press 1991.
- AN Shiryaev. Probability. Springer 1996.
The course material and much more is also available in the reference book:
- O Kallenberg. Foundations of Modern Probability. Springer 2021.
Substitutes for Courses
valid for whole curriculum period:
Prerequisites
valid for whole curriculum period:
FURTHER INFORMATION
Further Information
valid for whole curriculum period:
Teaching Language: English
Teaching Period: 2024-2025 Autumn I
2025-2026 Autumn I