Prerequisite: One compter language The class's material is divided in four blocks respectively dealing with Markov chains, continuous time Markov chains, Gaussian processes and stationary processes.
Although we use the word experiment more often than not we simulate the stochastic system in a computer and analyze the outcomes of these virtual experiments. These analysis can only take us so far and is usually complemented with numerical analysis of experimental outcomes. By theoretical analysis we refer to a set of tools which let us discover and understand properties of the system. With respect to analysis we distinguish between what we could call theoretical and experimental analysis.
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The goal of the class is to learn how to model, analyze and simulate stochastic systems. They also find application elsewhere, including social systems, markets, molecular biology and epidemiology. Stochastic systems are at the core of a number of disciplines in engineering, for example communication systems and machine learning. Stochastic systems analysis and simulation ( ESE 303) is a class that explores stochastic systems which we could loosely define as anything random that changes in time. The student will see surprises in election-day counting of ballots, a historical wager the sun will rise tomorrow, the folly of gambling, the sad news about lethal genes, the curiously persistent illusion of the hot hand in sports, the unreasonable efficacy of polls and its implications to medical testing, and a host of other beguiling settings.ĮSE 303 Stochastic Systems Analysis and Simulation But a bald listing of topics does not do justice to the subject: the material is presented in its lush and glorious historical context, the mathematical theory buttressed and made vivid by rich and beautiful applications drawn from the world around us.
The topics covered include: discrete and continuous probability spaces, distributions, mass functions, densities conditional probability independence the Bernoulli schema: the binomial, Poisson, and waiting time distributions uniform, exponential, normal, and related densities expectation, variance, moments conditional expectation generating functions, characteristic functions inequalities, tail bounds, and limit laws. The course begins with an exploration of combinatorial probabilities in the classical setting of games of chance, proceeds to the development of an axiomatic, fully mathematical theory of probability, and concludes with the discovery of the remarkable limit laws and the eminence grise of the classical theory, the central limit theorem. This course introduces students to the mathematical foundations of the theory of probability and its rich applications.
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A weekly lab accompanies the course where concepts discussed in class will be illustrated by hands-on projects students will be exposed to state-of-the-art test equipment and software tools (LabView, Spice). Applications include analog and digital circuits, such as single stage amplifiers and simple logic gates. It continues with introduction of non-linear elements such as diodes and MOSFET (MOS) transistors. It discusses the frequency behavior of circuits and the use of transfer functions. It continues with 1st order and 2nd order circuits in both the time and frequency domains. Today mathematical analysis is used to gain insight that supports design and more detailed and accurate representations of circuit performance are obtained using computer simulation. It starts with basic electric circuit analysis techniques of linear circuits. Designing, building and experimenting with electrical and electronic circuits are challenging and fun. This course gives an introduction of modern electric and electronic circuits and systems.
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