### Lecture 1: Basic principles of counting

##### Lecture Details :

Probability Theory and Applications by Prof. Prabha Sharma,Department of Mathematics,IIT Kanpur.For more details on NPTEL visit http://nptel.ac.in.

##### Course Description :

Basic principles of counting - Sample space , events, axioms of probability - Conditional probability, Independence of events - Random variables, cumulative density function, expected value - Discrete random variables and their distributions - Continuous random variables and their distributions - Function of random variables, Momement generating function - Jointly distributed random variables, Independent r. v. and their sums - Independent r. v. and their sums - Chi square r. v., sums of independent normal r. v., Conditional distr - Conditional distributions, Joint distr. of functions of r. v., Order statistics - Order statistics, Covariance and correlation - Covariance, Correlation, Cauchy- Schwarz inequalities, Conditional expectation - Conditional expectation, Best linear predictor - Inequalities and bounds - Convergence and limit theorems - Central limit theorem

Applications of central limit theorem - Strong law of large numbers, Joint mgf - Convolutions - Stochastic processes: Markov process - Transition and state probabilities - State prob., First passage and First return prob - First passage and First return prob. Classification of states - Random walk, periodic and null states - Reducible Markov chains - Time reversible Markov chains - Poisson Processes - Inter-arrival times, Properties of Poisson processes - Queuing Models: M/M/I, Birth and death process, Littles formulae - Analysis of L, Lq ,W and Wq , M/M/S model - M/M/S , M/M/I/K models - M/M/I/K and M/M/S/K models - Application to reliability theory failure law - Exponential failure law, Weibull law - Reliability of systems