Probability Theory (ΣΕΕ9)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΣΣΕ9
Semester	2
Course Title	Probability Theory
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Specialized general knowledge
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English, reading Course)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

This course treats the fundamentals of probability theory with a focus on proofs and rigorous mathematical theory. Upon its completion the students will be able to: explain the foundations of probability in the language of measure theory, state the strong law of large numbers have a working knowledge of weak convergence, characteristic functions, and the central limit theorem, explain the concept of conditional expectation, its properties and applications give an introduction to discrete time martingales and the martingale convergence theorem be able to solve basic problems related to the theory

General Competences

Working independently Decision-making Production of free, creative and inductive thinking

Syllabus

Measure-theoretic foundations of probability theory (σ-algebras, measure and probability spaces, generated sigma-algebras. Caratheodory extension theorem, Lebesgue measure, Random variables and their distribution Lebesgue integral and expectation. Almost sure convergence. Convergence in probability and in Lp. Monotone convergence theorem, dominated convergence theorem, Change of variables. Independent random variables). Key limit theorems (Weak law of large numbers, Borel-Cantelli lemmas, Kolmogorov extension theorem, strong law of large numbers, Lindeberg central limit theorem ) Martingales (Martingale Convergence, Applications)

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face-to-face)
Use of Information and Communications Technology	Use of ICT in communication with students
Teaching Methods	Activity Semester Workload Lectures 39 Working independently 78 Exercises - Homework 70.5 Course total 187.5
Student Performance Evaluation	Final written exam in Greek (in case of Erasmus students in English).

Attached Bibliography

• Billingsley P., Probability and Measure, 4th Edition, 1995, John Wiley and Sons
• M. Capinski and E. Kopp, Measure, integral and probability, Springer. (Springer-Verlag London, Ltd., second edition, 2004).
• R. Durrett, Probability: Theory and Examples, 4th Edition, Cambridge Series in Statistical and Probabilistic Mathematics, 2010.
• Kingman, J. F. C. and Taylor, S. J. An Introduction to Measure and Probability. Cambridge, England: Cambridge University Press, 1966.
• Rao, M. M. Measure Theory And Integration. New York: Wiley, 1987.
• D. Stroock, Probability: An Analytic View, 2nd Edition, Cambridge University Press, 2011
• [Περιοδικό / Journal] Advances in Applied Probability
• [Περιοδικό / Journal] Annals of Applied Probability
• [Περιοδικό / Journal] Annals of Probability
• [Περιοδικό / Journal] Journal of Applied Probability
• [Περιοδικό / Journal] Journal of Theoretical Probability
• [Περιοδικό / Journal] Probability Surveys
• [Περιοδικό / Journal] Theory of Probability and Its Applications