Course description

Infinite Coin Toss Model - Conditional Probability and Independence - Borel-Cantelli Lemmas,Random Variables : RVs as measurable functions - Probability law, types of RVs, and CDF - Multiple Random Variables and Independence - Jointly Continuous Random Variables,Conditional Distributions - Sums of Random Variables - General Transformations of Random Variables, Jacobian formula

Integration and Expectation : Abstract Integration - Properties of Abstract Integrals - Monotone Convergence Theorem - Integration over Different Spaces - Integration of Continuous Random Variables,Radon-Nikodym theorem - Fatou's Lemma and Dominated Convergence Theorem - Variance and Covariance - Conditional Expectation and MMSE estimate,Transforms : Probability Generating Functions - Moment Generating Functions - Characteristic Functions - Inversion Theorem and Uniqueness of the Inversion - Concentration Inequalities,Limit theorems : Convergence of Random Variables and related theorems - Weak Law of Large Numbers - Strong Law of Large Numbers - Central limit theorem, Multi-variate Gaussian Distribution

What will i learn?