Probability Theory and Applications

Overview

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Explore the fundamentals and advanced concepts of probability theory and its applications in this comprehensive 23-hour course. Delve into queuing models, Markov chains, stochastic processes, and reliability theory. Learn about birth and death processes, Little's formulae, and the strong law of large numbers. Study joint moment generating functions, reducible Markov chains, and Poisson processes. Examine the central limit theorem, random walks, and state classifications. Analyze various queuing models, including M/M/I, M/M/S, and M/M/I/K. Investigate convergence and limit theorems, transition probabilities, and time-reversible Markov chains. Explore reliability theory, including exponential and Weibull failure laws. Master discrete and continuous random variables, their distributions, and functions of random variables. Gain practical knowledge applicable to real-world scenarios in operations research, engineering, and data analysis.

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.