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Contents 1 Course Code 2 Credit 3 Prerequisites 4 Content/ Aims 5 Learning Outcomes 6 Suggested Textbooks 7 Outline

Course Code

IAM 541 (9700541)

Credit

(3-0) 3

Prerequisites

Advanced Calculus, consent of the instructor

Content/ Aims

The objective of this course is to initiate students to Probability Theory in which the main tools are those of Measure Theory. The proposed outline constitutes the prerequisites for Stochastic Calculus and other studies in the domain of stochastic processes.

The content of the course covers probability spaces, independence, conditional probability, product probability spaces, random variables and their distributions, distribution functions, mathematical expectation ( integration with respect to a probability measure), Lp-spaces, moments and generating functions, conditional expectation, linear estimation, Gaussian vectors, various convergence concepts, central limit theorem and laws of large numbers.

Learning Outcomes

Having followed this course, students should have knowledge of the formal basis for probability theory, and of the precise meaning of concepts such as random variable and expectation in a general setting. Furthermore; they should have the ability to model problems by means of the language of formal probability theory and the ability to learn in unfamiliar contexts, and to express arguments clearly and accurately.

Suggested Textbooks

• Elements of Probability Theory, Hayri Körezlioğlu and Azize Bastıyalı Hayfavi, ODTU, 20001
• Probability Essentials, J. Jacod and Ph. Protter, Springer, 2000
• An Introduction to Measure and Probability,J.C. Taylor, Springer, 1997
• An Introduction to Probability Theory and its Applications,W. Feller, vol.1 + vol.2

Outline

• Probability spaces : sigma-algebra, probability measure, independence of events, conditional probability, Bayes Formula,(2 weeks).
• Random variables, distribution functions, product probability spaces, independence of random variables,(2 weeks).
• Expectation (integration) of random variables. Monotone Convergence Theorem, Dominated Convergence Theorem, (2 weeks).
• Lp-spaces. Almost sure convergence, convergence in probability, convergence in Lp. Uniform integrability, (2 weeks).
• Moments. Laplace and Fourier transforms, (1week).
• Conditional expectation, (2 weeks).
• Linear estimation and Gaussian vectors, (1 week).
• Convergence in distribution, Central Limit Theorem, Laws of large numbers, (2 weeks).