## 2018-03-16

### Probabilistic Analysis (Fall 2017) by Çağın Ararat at Bilkent University

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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