Probability Theory

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

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.

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

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.

Central limit theorem for a sequence of independent random variables under Linderberg’s condition, central limit theorem of i.i.d. with finite variance, sequence of events and random variables, Zero-One law of Borel and Kolmogorov almost sure convergence in mean square, Kitchin's weak law of large numbers and strong law of large numbers.