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Probability Theory and Applications

Probability Theory and Applications. Instructor: Prof. Prabha Sharma, Department of Mathematics and Statistics, IIT Kanpur. The aim of this course is to familiarise students with basic concepts of Probability Theory with reasonable amount of rigor. Examples will be used for motivating students to learn and understand the basic concepts. Topics include: basics of probability theory, random variables, moments and other functions of random variables, limit theorems and inequalities, Poisson process, and Markov chains. (from nptel.ac.in)

Basic Principles of Counting


Lecture 01 - Basic Principles of Counting
Lecture 02 - Sample Space, Events, Axioms of Probability
Lecture 03 - Conditional Probability, Independence of Events
Lecture 04 - Random Variables, Cumulative Density Function, Expected Value
Lecture 05 - Discrete Random Variables and their Distributions
Lecture 06 - Discrete Random Variables and their Distributions (cont.)
Lecture 07 - Discrete Random Variables and their Distributions (cont.)
Lecture 08 - Continuous Random Variables and their Distributions
Lecture 09 - Continuous Random Variables and their Distributions (cont.)
Lecture 10 - Continuous Random Variables and their Distributions (cont.)
Lecture 11 - Function of Random Variables, Moment Generating Function
Lecture 12 - Jointly Distributed Random Variables, Independent Random Variables and their Sums
Lecture 13 - Independent Random Variables and their Sums
Lecture 14 - Chi-Square Random Variables, Sums of Independent Normal R. V., Conditional Distribution
Lecture 15 - Conditional Distribution, Joint Distribution of Functions of Random Variables, Order Statistics
Lecture 16 - Order statistics, Covariance and Correlation
Lecture 17 - Covariance, Correlation, Cauchy-Schwarz Inequalities, Conditional Expectation
Lecture 18 - Conditional Expectation, Best Predictor
Lecture 19 - Inequalities and Bounds
Lecture 20 - Convergence and Limit Theorem
Lecture 21 - Central Limit Theorem
Lecture 22 - Applications of Central Limit Theorem
Lecture 23 - Strong Law of Large Numbers, Joint Moment Generating Function
Lecture 24 - Convolutions
Lecture 25 - Stochastic Processes: Markov Process
Lecture 26 - Transition and State Probabilities
Lecture 27 - Steady State Probabilities, First Passage and First Return Probabilities
Lecture 28 - First Passage and First Return Probabilities, Classification of States
Lecture 29 - Random Walk, Periodic and Null States
Lecture 30 - Reducible Markov Chains
Lecture 31 - Time Reversible Markov Chains
Lecture 32 - Poisson Processes
Lecture 33 - Inter-Arrival Times, Properties of Poisson Processes
Lecture 34 - Queuing Models: M/M/1, Birth and Death Process, Little's Formulae
Lecture 35 - Analysis of L, Lq, W and Wq, M/M/S Model
Lecture 36 - M/M/S , M/M/1/K Models
Lecture 37 - M/M/1/K and M/M/S/K Models
Lecture 38 - Application to Reliability Theory, Failure Law
Lecture 39 - Exponential Failure Law, Weibull Law
Lecture 40 - Reliability of Systems

References
Probability Theory and Applications
Instructor: Prof. Prabha Sharma, Department of Mathematics and Statistics, IIT Kanpur. The aim of this course is to familiarise students with basic concepts of Probability Theory with reasonable amount of rigor.