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source: BilkentUniversitesi 2017年9月22日

IE 523 Probabilistic Analysis

Department of Industrial Engineering

Axiomatic construction of probability theory, properties of probability, conditional probability, independence. Discrete and continuous random variables and vectors (distribution function, expectation, variance, moments). Chebyshev inequality and law of large numbers. Conditional expectation. Transformations of random variables. Generating and characteristics functions. Asymptotic methods in probability theory, types of convergence of random variables. Sums of independence random variables, central limit theorem, Poisson theorem. Selected topics.

Lecture 01: Introduction, Random Experiment

Lecture 02: Sample Space, Event

Lecture 03: Sigma Algebra, Generated Sigma Algebra

Lecture 04: Borel Sigma Algebra, Trace

Lecture 05: Monotone Class Theorem

Lecture 06: Probability Measure

Lecture 07: Probability Measure, Lebesgue Measure

Lecture 08: Measure Space

Lecture 09: Random Variables, Measurable Functions

Lecture 10: Operations with Random Variables

Lecture 11: Limits of Random Variables

Lecture 12: Monotone Class Theorem for Functions

Lecture 13: Expectations and Lebesgue Integrals

Lecture 14: Monotone Convergence Theorem

Lecture 15: Limit Theorems for Integrals

Lecture 16: Examples of Integrals, Insensitivity

Lecture 17: Expectation, Distribution

Lecture 18: Radon - Nikodym Theorem

Lecture 19: Discrete Random Variables

Lecture 20: Continuous Random Variables

Lecture 21: Laplace Transform

Lecture 22: Fourier Transform

Lecture 23: Existence of Random Variables

Lecture 24: Product Spaces

Lecture 25: Measurability in Product Spaces

Lecture 26: Measures on Product Spaces

Lecture 27: Measures on Product Spaces

Lecture 28: Product of a Kernel and a Measure

Lecture 29: Fubini and Tonelli Theorems

Lecture 30: Joint Distributions

Lecture 31: Independence

Lecture 32: Joint Distributions and Independence

Lecture 33: Gaussian Random Vectors

Lecture 34: Conditional Expectation

Lecture 35: Conditional Determinism, Tower Property

Lecture 36: Conditional Expectation given A Random Variable

Lecture 37: Regular Conditional Probability

Lecture 38: Conditional Distributions

Lecture 39: Conditioning Examples

Lecture 40: Introduction to Stochastic Processes

Lecture 41: Ionescu - Tulcea Theorem

Lecture 42: Bernoulli Processes

Lecture 43: Almost Sure Convergence

Lecture 44: Convergence in Probability

Lecture 45: Lp Spaces

Lecture 46: Convergence in Lp

Lecture 47: Law of Large Numbers, Central Limit Theorem

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