# Probability Theory (8 ECTS)

Course Code:
6116
Semester:
6th
Elective Courses
Διδάσκων:

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