Preliminaries : Set Theory – Real Analysis basics – Cardinality and Countability,Probability Measures : Probability Spaces – Properties of Probability Measures – Discrete Probability Spaces – Borel Sets and Lebesgue Measure – 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

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Course Curriculum

INTRODUCTION Details 34:55
CARDINALITY AND COUNTABILITY-1 Details 41:53
CARDINALITY AND COUNTABILITY-2 Details 39:34
PROBABILITY SPACES-1 Details 52:11
PROBABILITY SPACES-2 Details 51:30
PROPERTIES OF PROBABILITY MEASURES Details 50:6
DISCRETE PROBABILITY SPACES Details 45:40
GENERATED Σ-ALGEBRA, BOREL SETS Details 27:19
BOREL SETS AND LEBESGUE MEASURE-1 Details 50:11
BOREL SETS AND LEBESGUE MEASURE-2 Details 50:27
THE INFINITE COIN TOSS MODEL Details 49:24
CONDITIONAL PROBABILITY AND INDEPENDENCE Details 50:22
INDEPENDENCE CONTD. Details 41:5
THE BOREL-CANTELLI LEMMAS Details 51:11
RANDOM VARIABLES Details 47:41
CUMULATIVE DISTRIBUTION FUNCTION Details 46:16
TYPES OF RANDOM VARIABLES Details 45:17
CONTINUOUS RANDOM VARIABLES Details 44:33
CONTINUOUS RANDOM VARIABLES (CONTD.) AND SINGULAR RANDOM VARIABLES Details 47:20
SEVERAL RANDOM VARIABLES Details 49:29
INDEPENDENT RANDOM VARIABLES-1 Details 46:10
INDEPENDENT RANDOM VARIABES-2 Details 47:8
JOINTLY CONTINUOUS RANDOM VARIABLES Details 46:9
TRANSFORMATION OF RANDOM VARIABLES-1 Details 52:14
TRANSFORMATION OF RANDOM VARIABLES-2 Details 46:45
TRANSFORMATION OF RANDOM VARIABLES-3 Details 44:38
TRANSFORMATION OF RANDOM VARIABLES-4 Details 48:20
INTEGRATION AND EXPECTATION-1 Details 50:21
INTEGRATION AND EXPECTATION-2 Details 43:5
PROPERTIES OF INTEGRALS Details 50:28
MONOTONE CONVERGENCE THEOREM Details 47:26
EXPECTATION OF DICRETE RANDOM VARIABLES, EXPECTATION OVER DIFFERENT SPACES Details 47:29
EXPECTATION OF DICRETE RANDOM VARIABLES Details 46:17
FATOU’S LEMMA & DOMINATED CONVERGENCE THEOREM Details 42:9
VARIANCE AND COVARIANCE Details 49:1
COVARIANCE, CORRELATION COEFFICIENT Details 42:56
CONDITIONAL EXPECTATION Details 53:28
MMSE ESTIMATOR, TRANSFORMS Details 44:54
MOMENT GENERATING FUNCTION Details 50:22
CHARACTERISTIC FUNCTION – 1 Details 49:21
CHARACTERISTIC FUNCTION – 2 Details 43:39
CONCENTRATION INEQUALITIES Details 0:47
CONVERGENCE OF RANDOM VARIABLES – 1 Details 43:49
CONVERGENCE OF RANDOM VARIABLES – 2 Details 50:37
CONVERGENCE OF RANDOM VARIABLES – 3 Details 45:51
CONVERGENCE OF CHARCTERISTIC FUNCTIONS, LIMIT THEOREMS Details 45:18
THE LAWS OF LARGE NUMBERS Details 49:7
THE CENTRAL LIMIT THEOREM Details 43:29
A BRIEF OVERVIEW OF MULTIVARIATE GAUSSIANS Details 58:54

This course is delivered by NPTEL, is part of IIT Madras online courses.

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