Probability spaces, probability measures on countable and uncountable spaces, conditional probability, independence; Random variables and vectors, distribution functions, functions of random vectors, standard univariate and multivariate discrete and continuous distributions and their properties; Mathematical expectations, moments, moment generating functions, characteristic functions, inequalities, conditional expectations, covariance, correlation; Modes of convergence of sequences of random variables, weak and strong laws of large numbers, central limit theorems; Introduction to stochastic processes, definitions and examples, Markov chains, Poisson processes, Brownian motion.

Texts/References:

• J. Jacod and P. Protter, Probability Essentials, Springer, 2004.
• S. Ross, Introduction to Probability Models, 10th Edition, Elsevier, 2010. [Newer Editions are also available]
• G. R. Grimmett and D. R. Stirzaker, Probability and Random Processes, 3rd edition, Oxford University Press, 2001.
• V. K. Rohatgi and A. K. Md. E. Saleh, An Introduction to Probability and Statistics, 2nd edition, Wiley, 2001.
• J. Rosenthal, A First Look at Rigorous Probability Theory, 2nd edition, World Scientific, 2006.
• P. G. Hoel, S. C. Port and C. J. Stone, Introduction to Probability Theory, Universal Book Stall, 2000.
• W. Feller, An Introduction to Probability Theory and its Applications, Vol. 1, 3rd Edition, Wiley, 1968.