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# Probability Theory and Applications

### Overview

This course covers a wide range of topics in Probability Theory and Applications. The learning outcomes include understanding queuing models, Markov chains, Poisson processes, central limit theorem, stochastic processes, reliability theory, and various distributions of random variables. Students will learn to analyze and apply these concepts in real-world scenarios. The course teaches skills such as calculating transition probabilities, analyzing state probabilities, and understanding the properties of different types of random variables. The teaching method involves theoretical explanations, mathematical derivations, and practical applications. This course is intended for individuals interested in deepening their knowledge of probability theory and its applications, particularly students and professionals in fields such as mathematics, statistics, engineering, and computer science.

### Syllabus

Queuing Models M/M/I Birth and Death Process Little's Formulae.
Strong law of large numbers, Joint mgf.
Reducible markov chains.
Inter-arrival times, Properties of Poisson processes.
Applications of central limit theorem.
Random walk, periodic and null states.
Poisson processes.
Central limit theorem.
First passage and first return prob. Classification of states.
Convergence and limit theorems.
State prob.First passage and First return prob.
M/M/I/K & M/M/S/K Models.
Inequalities and bounds.
Transition and state probabilities.
M/M/S M/M/I/K Model.
Stochastic processes:Markov process.
Convolutions.
Time Reversible Markov Chains.
Analysis of L,Lq,W and Wq, M/M/S Model.
Reliability of systems.
Exponential Failure law, Weibull Law.
Application to Reliability theory failure law.
Function of Random variables,moment generating function.
Continuous random variables and their distributions.
Continuous random variables and their distributions.
Discreet random variables and their distributions.
Discreet random variables and their distributions.
Discrete random variables and their distributions.

### Taught by

Ch 30 NIOS: Gyanamrit

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