3 Credits (3-0-0)

Axiomatic definition of a probability measure, examples, properties of the probability measure, finite probability space, conditional probability and Bayes formula, countable probability space, general probability space.

Random variables, examples, sigma-field generated by a random variable, tail sigma-field, probability space on R induced by a random variable. Independent events, sigma-fields and random variables, Borel 0-1 criteria, Kolmogorov 0-1 criteria.

Distribution - definition and examples, properties, characterization, Jordan decomposition theorem, discrete, continuous and mixed random variables, standard discrete and continuous distributions, convolution of distributions.

Two dimension random variables, joint distributions, marginal distributions, operations on random variables and their corresponding distributions, multidimensional random variables and their distributions.

Expectation of a random variable, expectation of a discrete and a continuous random variable, moments and moment generating function, correlation, covariance and regression.

Various modes of convergence, Weak law of large numbers, strong law of large numbers.

Convergence in distribution, weak convergence of generalized distributions, Helly-Bray theorems, Scheffe’s theorem.

Characteristic function – definition and examples, properties, uniqueness and inversion theorems, moments using characteristic function, Paul Levy’s continuity property of characteristic functions, characterization of independent random variables.

Central limit theorem – Liapunov’s and Lindberg’s condition, LindebergLevy form.

Infinite divisibility, Levy-Khintchine theorem.