### Probability and Statistics (Video) - Mathematics Free Video Lectures

 COURSE OUTLINE Algebra of sets, introduction to probability, random variables, probability distributions, moments, moment generating function, Markov and Chebyshev inequalities, special discrete and continuous distributions, function of a random variable, joint distributions, bivariate normal distribution, transformation of random vectors, central limit theorem, sampling distributions, point estimation, unbiasedness, consistency, method of moments and maximum likelihood estimation, confidence intervals for parameters in one sample and two sample problems from normal populations, testing of hypotheses, Neyman-Pearson lemma, tests for one sample and two sample problems for normal populations.

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 Modules Topics and Contents Number of Lectures 1. Introduction Preliminaries Existence, Uniqueness, and Wellposedness Stability and Asymptotic Stability 3 2. Single Step Methods The Euler Method Convergence of Euler’s Method Improvement of the error bound Stability 4 3. Higher order Single Step Methods Higher Order Methods Runge-Kutta Methods Error bounds for Runge-Kutta methods Absolute Stability for Runge-Kutta Methods 4 4. Systems of Equations and Equations of Order Greater Than One Systems of Equations Direct Methods For Higher Order Equations 2 5. Consistency, Stability and Convergence of General Single – Step Methods General Single Step Methods Convergence of General One-Step Methods 2 6. Implicit Runge-Kutta Methods Derivation of Implicit Runge-Kutta methods Derivation of Implicit Runge-Kutta Methods(Contd.) 2 7. Multistep Methods Multistep Methods Multistep Methods (Contd.) Multistep Methods(Contd.) The local error of the formulas based on integration Local Error of Nystrom & Milne-Simpson Methods Multistep Methods for Special Equations of the Second Order Special 2nd order equations(Contd.) 7 8. Linear Multistep Methods Linear Multistep Methods Linear Multistep Methods (Contd) Consistency and Zero-Stability of Linear Multistep Methods Convergence of Linear Multistep Methods Necessary & Sufficient Conditions for Convergence Absolute Stability and Relative Stability General methods for finding intervals of absolute and relative stability Some more methods for Absolute & Relative Stability 8 9. Stiff-Initial Value Systems First order linear systems with constant coefficient Stiffness and Problem of Stiffness The problem of implicitness for Stiff systems Linear multistep methods for Stiff systems 4 10. Finite Difference Methods for Boundary Value Problems Finite Difference Methods Analysis of Difference System Analytic Expressionof the Error Nonlinear second order equations Special Boundary Value Problems Special Boundary Value Problems(Contd) 6 Total number of lectures 42

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