Probability Theory (8 ECTS)

Course Code: 
Elective Courses

Non-countable sets and the necessity for axiomatic basis of probability spaces (σ-algebra events, Kolmogorov axiom, probability measure properties). The Lebesque-Caratheodory theorem (summarized, applications). Defining random variables and Borel countability. Stochastic independence, Borel- Canteli lemma, 0-1 Kolmogorov law. Random variable expected value as a probability measure and as a Lebesque integral, regarding the corresponding probability distribution on the Borel line, expected values properties. Types of random variables series convergence (almost certain, by b-class mean value, by probability, by distribution). Limit theorems (monotone convergence, Fatou lemma, dominated or bounded convergence theorem, uniform integrability, weak and strong Laws of the large numbers, Central Limit Theorem). Disconnection of the general probability distribution in its Lebesque components (discrete, continuous, unique continuous). The Radon-Nikodym theorem. Conditional expected value, conditional probability and its properties

       Recommended Reading:

  • Athreya, Krishna B., Lahiri, Soumendra N., Measure Thery and Probability Thery, Springer Science and Business Media, LLC, 2006.
  • Billingsley, P. (1995): Probability and Measure, 3rd Edition, John Wiley & Sons.
  • Bhattacharya, Rabi. Waymire, Edward C., A Basic Course on Probability Theory, Springer Science and Business Media, Inc., 2007.
  • Rosenthal, J. S. (2006): A First Look at Rigorous Probability Theory, Second Εdition, World Scientific.
  • Roussas, G.G. (2005): An Introduction to Measure-Theoretic Probability, Elsevier Academic Press.
  • Skorokhond, A.V., Prokhorov, Yu.V., Basic Principles and Applications of Probability Theory, Springer-Verlag Berlin Heidelberg, 2005.
  • SpringerLink (Online service), Gut A., Probability: Α graduate Course, Springer Science and Business Media, Inc., 2005.
  • Ρούσσας, Γ. Γ. (1992): Θεωρία Πιθανοτήτων, Eκδόσεις ΖΗΤΗ, Θεσσαλονίκη.
  • Καλπαζίδου, Σ. (2002): Στοιχεία Μετροθεωρίας Πιθανοτήτων, Eκδόσεις ΖΗΤΗ, Θεσσαλονίκη.