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Probability Concepts

Probability Models. Probability Models.

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Probability Concepts

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  1. Jayendran Venkateswaran IE & OR IIT Bombay IE 502: Probabilistic Models

  2. What is the course about? • You will learn how probability theory can be applied to study (model/ analyze/ understand) problems in fields such as engineering, management, operations research, medicine, computer science, and social systems. • “Probability theory is the branch of mathematics concerned with analysis of random phenomena” • Encyclopedia Britannica : Random phenomena  Uncertainty • Probability theory and Stochastic models are quite useful in modeling real-life situations

  3. History • 16th century: Gerolamo Cardano • Father of Probability • Computed ‘chances’ of dice throws • 17th century: Fermat & Pascal • Joint founders of Probability Theory • “Problem of Points”, computed fair odds in games of chance • Ideas were further popularized by Huygens

  4. History (contd.) • 18th Century: Bernoulli • James, Nicholas, Daniel • Observed frequency as probability; theorems and rules for combining probability, law of large numbers, Bernoulli trials, etc • De Moivre’s “Doctrine of Chance”, introduced Normal distribution; applied to actuarial/ mortality tables.

  5. History (contd.) • Late 18th - 19th Century: Laplace & Gauss • Use of probability in data analysis, mainly astronomical data • Inverse Probability, Bayes’ theorem, least squares, Normal distribution, generating functions, central limit theorem … • “Théorie analytique des probabilités”

  6. History (contd.) • 20th Century • Fisher & Neyman introduced hypothesis testing  closely connected probability & statistics • Notions of stochastic processes; Morkov process; Brownian motion • Kolmogorov’s “Basic notions of the calculus of probability”  axioms of probability  measure theory

  7. Course Topics • Probability Theory • Random Variables • Conditional Probability & Expectations • Exponential distribution & related models • Reliability Theory • Markov Chains • Elementary Queueing Theory • Other applications

  8. Resources - Books • Introduction to Probability Models Sheldon Ross, 9th edition, Academic Press • Intro to Probability Models: Operations Research W.L. Winston, 4th edition, Duxbury Resource Center • Operations Research H.A. Taha, 8th edition, Prentice Hall • Introduction to Probability D.P. Bertsekas and J.N. Tsitsiklis 2nd edition, Athena Scientific

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