Code MATH 524
Term 201802
Title Probability Theory
Faculty Faculty of Engineering and Natural Sciences
Subject Mathematics(MATH)
SU Credit 3
ECTS Credit 10.00
Instructor(s) Turgay Bayraktar tbayraktar@sabanciuniv.edu,
Detailed Syllabus
Language of Instruction English
Level of Course Doctoral
Master
Prerequisites
(only for SU students)
--
Mode of Delivery Formal lecture
Planned Learning Activities Interactive,Learner centered,Discussion based learning,Task based learning
Content

Semi-algebras and sigma-algebras of events, Kolmogorov?s axioms of probability, consequences thereof, probability spaces, measurability, random variables as measurable mappings, random vectors, probability measures induced on Borel sigma-algebras by random vectors, distributions and distribution functions, extension of probability measure starting by semi-algebras, mathematical expectation, expected values of non-negative simple, non-negative and general random variables, properties, conditional distributions and independence, Borel-Cantelli lemma, conditional expectation given a sub sigma-algebra, Radon-Nikodym theorem, different modes of convergence, almost sure convergence, convergence in probability, convergence in L^p, convergence in distribution, different implications between them, characteristic functions, inversion formulas, relation to convergence concepts, the weak and the strong law of large numbers, central limit theorem.

Learning Outcome

Upon completing this course students should be able to:

1. define axiomatic foundations of the probability theory
2. apply measure-theoretic approach to the important concepts and the proofs of fundamental results
3. use sigma algebras, fields and semialgebras of events
4. use measurability concepts as related to the random variables and random vectors
5. analyze expectations as abstract Lebesgue integrals constructed on the outcome spaces
6. prove Integral convergence theorems
7. evaluate different modes of convergences of sequences of random variables
8. identify characteristic functions and their relations to convergence in distributions

Programme Outcomes