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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,
Detailed Syllabus
Language of Instruction English
Level of Course Doctoral
Type of Course Click here to view.
(only for SU students)
Mode of Delivery Formal lecture
Planned Learning Activities Interactive,Learner centered,Discussion based learning,Task based learning

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
Common Outcomes For All Programs
1 Develop and deepen the current and advanced knowledge in the field with original thought and/or research and come up with innovative definitions based on Master's degree qualifications 4
2 Conceive the interdisciplinary interaction which the field is related with ; come up with original solutions by using knowledge requiring proficiency on analysis, synthesis and assessment of new and complex ideas. 4
3 Evaluate and use new information within the field in a systematic approach. 4
4 Develop an innovative knowledge, method, design and/or practice or adapt an already known knowledge, method, design and/or practice to another field; research, conceive, design, adapt and implement an original subject. 4
5 Critical analysis, synthesis and evaluation of new and complex ideas. 4
6 Gain advanced level skills in the use of research methods in the field of study. 4
7 Contribute the progression in the field by producing an innovative idea, skill, design and/or practice or by adapting an already known idea, skill, design, and/or practice to a different field independently. 3
8 Broaden the borders of the knowledge in the field by producing or interpreting an original work or publishing at least one scientific paper in the field in national and/or international refereed journals. 4
9 Demonstrate leadership in contexts requiring innovative and interdisciplinary problem solving. 3
10 Develop new ideas and methods in the field by using high level mental processes such as creative and critical thinking, problem solving and decision making. 4
11 Investigate and improve social connections and their conducting norms and manage the actions to change them when necessary. 4
12 Defend original views when exchanging ideas in the field with professionals and communicate effectively by showing competence in the field. 4
13 Ability to communicate and discuss orally, in written and visually with peers by using a foreign language at least at a level of European Language Portfolio C1 General Level. 4
14 Contribute to the transition of the community to an information society and its sustainability process by introducing scientific, technological, social or cultural improvements. 4
15 Demonstrate functional interaction by using strategic decision making processes in solving problems encountered in the field. 4
16 Contribute to the solution finding process regarding social, scientific, cultural and ethical problems in the field and support the development of these values. 3
Common Outcomes For All Programs
1 Develop the ability to use critical, analytical, and reflective thinking and reasoning 5
2 Reflect on social and ethical responsibilities in his/her professional life. 4
3 Gain experience and confidence in the dissemination of project/research outputs 4
4 Work responsibly and creatively as an individual or as a member or leader of a team and in multidisciplinary environments. 4
5 Communicate effectively by oral, written, graphical and technological means and have competency in English. 3
6 Independently reach and acquire information, and develop appreciation of the need for continuously learning and updating. 4
Common Outcomes ForFaculty of Eng. & Natural Sci.
1 Design and model engineering systems and processes and solve engineering problems with an innovative approach. 2
2 Establish experimental setups, conduct experiments and/or simulations. 2
3 Analytically acquire and interpret data. 3
Physics (non-thesis) Program Outcomes Core Electives
1 Employ mathematical methods to solve physical problems and understand relevant numerical techniques. 4
2 Conduct basic experiments or simulations. 2
3 Analytically acquire and interpret data. 3
4 Establish thorough understanding of the fundamental principles of physics. 3
Electronics Engineering (with thesis) Program Outcomes Core Electives
1 Use advanced Math (including probability and/or statistics), advanced sciences, advanced computer and programming, and advanced Electronics engineering knowledge to design and analyze complex electronics circuits, instruments, software and electronic systems with hardware/software. 5
2 Analyze and design advanced communication networks and systems, advanced signal processing algorithms or software using advanced knowledge on diff. equations, linear algebra, complex variables and discrete math. 2
Mechatronics Engineering (with thesis) Program Outcomes Area Electives
1 Apply software, modeling, instrumentation, and experimental techniques and their combinations in the design and integration of electrical, electronic, control and mechanical systems. 3
2 Interact with researchers from different disciplines to exchange ideas and identify areas of research collaboration to advance the frontiers of present knowledge and technology; determine relevant solution approaches and apply them by preparing a research strategy. 4
3 Take part in ambitious and highly challenging research to generate value for both the industry and society. 2
Energy Technologies and Management (with thesis) Program Outcomes Area Electives
1 Assess and identify developments, strategies, opportunities and problems in energy security and energy technologies. 2
2 Define and solve technical, economic and administrative problems in energy businesses. 2
3 Establish knowledge and understanding of energy security, energy technologies, energy markets and strategic planning in energy enterprises. 2
4 Demonstrate an awareness of environmental concerns and their importance in developing engineering solutions and new technologies. 2
5 Acquire a series of social and technical proficiencies for project management and leadership skills. 2
Molecular Biology, Genetics and Bioengineering (with thesis) Program Outcomes Area Electives
1 Apply knowledge of key concepts in biology, with an emphasis on molecular genetics, biochemistry and molecular and cell biology. 3
2 Display an awareness of the contemporary biological issues in relation with other scientific areas. 2
3 Demonstrate hands-on experience in a wide range of biological experimental techniques. 2
Industrial Engineering (with thesis) Program Outcomes Area Electives
1 Establish a strong theoretical background in several of a broad range of subjects related to the discipline, such as manufacturing processes, service systems design and operation, production planning and control, modeling and optimization, stochastics, statistics. 5
2 Develop novel modeling and / or analytical solution strategies for problems in integrated production and service systems involving human capital, materials, information, equipment, and energy, also using an interdisciplinary approach whenever appropriate. 3
3 Implement solution strategies on a computer platform for decision-support purposes by employing effective computational and experimental tools. 2
4 Acquire skills to independently explore and tackle problems related to the discipline that were not encountered previously. Develop appropriate modeling, solution, implementation strategies, and assess the quality of the outcome. 3
Computer Science and Engineering (with thesis) Program Outcomes Area Electives
1 Apply knowledge of mathematics, science, and engineering in computer science and engineering related problems. 5
2 Display knowledge of contemporary issues in computer science and engineering and apply to a particular problem. 3
3 Demonstrate the use of results from interpreted data to improve the quality of research or a product in computer science and engineering. 3
Materials Science and Engineering (with thesis) Program Outcomes Area Electives
1 Apply a broad knowledge of structure & microstructure of all classes of materials, and the ability to use this knowledge to determine the material properties. 2
2 Apply a broad understanding of the relationships between material properties, performance and processing. 2
3 Apply a broad understanding of thermodynamics, kinetics, transport phenomena, phase transformations and materials aspects of advanced technology. 2
4 Demonstrate hands-on experience using a wide range of materials characterization techniques. 2
5 Demonstrate the use of results from interpreted data to improve the quality of research, a product, or a product in materials science and engineering. 2
Assessment Methods and Criteria
  Percentage (%)
Midterm 40
Homework 60
Recommended or Required Reading

Probability and Measure by Patrick Billingsley, Volume 245 of Wiley Series in Probability and Statistics 3rd ed.

Supplementary texts: Probability and Stochastics by Erhan C?nlar, Volume 261 of Springer Graduate Texts in Mathematics.