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
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