# Probability Theory and Applications

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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, Little’s 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

### Course Curriculum

 Basic principles of counting Details 52:47 Sample space , events, axioms of probability Details 52:17 Conditional probability, Independence of events. Details 52:38 Random variables, cumulative density function, expected value Details 48:14 Discrete random variables and their distributions Details 44:47 Discrete random variables and their distributions I Details 48:4 Discrete random variables and their distributions II Details 49:45 Continuous random variables and their distributions. Details 46:59 Continuous random variables and their distributions I Details 53:25 Continuous random variables and their distributions II Details 1:4:11 Function of random variables, Momement generating function Details 49:40 Jointly distributed random variables, Independent r. v. and their sums Details 51:12 Independent r. v. and their sums. Details 49:4 Chi â€“ square r. v., sums of independent normal r. v., Conditional distr. Details 40:53 Conditional disti, Joint distr. of functions of r. v., Order statistics Details 44:45 Order statistics, Covariance and correlation Details 47:31 Covariance, Correlation, Cauchy- Schwarz inequalities, Conditional expectation. Details 50:35 Conditional expectation, Best linear predictor Details 51:52 Inequalities and bounds. Details 45:16 Convergence and limit theorems Details 1:51 Central limit theorem Details 51:57 Applications of central limit theorem Details 48:5 Strong law of large numbers, Joint mgf. Details 40:38 Convolutions Details 47:20 Stochastic processes: Markov process. Details 42:46 Transition and state probabilities. Details 45:50 State prob., First passage and First return prob Details 46:53 First passage and First return prob. Classification of states. Details 47:43 Random walk, periodic and null states. Details 40:57 Reducible Markov chains Details 57:39 Time reversible Markov chains Details 52:44 Poisson Processes Details 53:13 Inter-arrival times, Properties of Poisson processes Details 46:32 Queuing Models: M/M/I, Birth and death process, Littleâ€™s formulae Details 50:11 Analysis of L, Lq ,W and Wq , M/M/S model Details 44:25 M/M/S , M/M/I/K models Details 52:16 M/M/I/K and M/M/S/K models Details 38:2 Application to reliability theory failure law Details 48:34 Exponential failure law, Weibull law Details 41:49 Reliability of systems Details 44:46

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