Probability Theory

Unit-I                                                                                                                

General probability space, various definition of probability, combinations of events, additive and multiplicative law of probability, conditional probability, Bayes’ theorem and its application.

Unit –II                                                                                                            

Concept of random variable, cumulative distribution function, probability distribution function, joint probability distribution function, marginal distribution function and their application, conditional distribution function and conditional probability distribution function of random variables and their distributions using: jacobian transformation, cumulative distribution function, moment generating function.

Unit-III                                                                                                             

Mathematical Expectation, moments, Sheppard’s correction, conditional expectation, moment generating function and their applications, cumulant generating function and their applications, characteristic function and its applications. Inversion Theorem, Continuity Theorem, Uniqueness Theorem.

Unit-IV                                                                                                               

 Levy’s continuity theorem (statement only), probabilities inequalities and their applications, Chebychev inequality, Markov and Jenson inequality. Convergence in probability and convergence in distribution, weak law of large numbers.

Unit-V                                                                                                           

Central limit theorem: De-Moivre’s Laplace, Liaponouff, Lindeberg-Levy and their simple problems, Zero-One law of Borel and Kolmogorov almost sure convergence in mean square, strong law of large numbers.