Probability and Random Models (ELEN90054)

AIMS

This subject provides an introduction to probability theory, random variables, random vectors, decision tests, and stochastic processes. Uncertainty is inevitable in real engineering systems, and the laws of probability offer a powerful way to evaluate uncertainty, to predict and to make decisions according to well-defined, quantitative principles. The material covered is important in fields such as communications, data networks, signal processing and electronics. This subject is a core requirement in the Master of Engineering (Electrical, Mechanical and Mechatronics).

INDICATIVE CONTENT

Topics include:

• Foundations – combinatorial analysis, axioms of probability, independence, conditional probability, Bayes’ rule;
• Random variables (rv’s)– definition; cumulative distribution, probability mass and probability density functions; expectation and variance; functions of an rv; important distributions and their properties and uses;
• Multiple random variables – joint cumulative distribution, probability mass and probability density functions; independent rv’s; correlation and covariance; conditional distributions and expectation; functions of several rv’s; jointly Gaussian rv’s; random vectors;
• Sums, inequalities and limit theorems – sums of rv’s, moment generating function; Markov and Chebychev inequalities; weak and strong laws of large numbers; the Central Limit Theorem;
• Decision testing - maximum likelihood, maximum a posterior, minimum cost and Neyman-Pearson rules; basic minimum mean-square error estimation;
• Stochastic processes – mean and autocorrelation functions, strict and wide-sense stationarity; ergodicity; important processes and their properties and uses;
• Introduction to Markov chains.

This material is complemented by exposure to examples from electrical engineering and software tools (e.g. MATLAB) for computation and simulations.