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Random Signals and Systems

This course will introduce the student to the fundamental concepts of probability theory applied to engineering problems. Its goal is to develop the ability to construct and exploit probabilistic models in a manner that combines intuition and mathematical precision. The proposed treatment of probability includes elementary set operations, sample spaces and probability laws, conditional probability, independence, and notions of combinatorics. A discussion of discrete and continuous random variables, common distributions, functions, and expectations forms an important part of this course. Transform methods, limit theorems, modes of convergence, and bounding techniques are also covered. In particular, special consideration is given to the law of large numbers and the central limit theorem. Many examples from engineering, science, and statistics will be provided.

Major Goals

  1. Review basic notions of set theory and simple operations such as unions, intersections, differences and De Morgan's laws. Discuss cartesian products and simple combinatorics. Go over the counting principle, permutations, combinations and partitions.
  2. Introduce sample spaces, probability laws and random variables. Distinguish between events and outcomes, and illustrate how to compute their probabilities.
  3. Present the concepts of independence and conditional probabilities. Study the total probability theorem and Bayes' rule. Provide examples of these important results applied to tangible engineering problems.
  4. Understand mathematical descriptions of random variables including probability mass functions, cumulative distribution functions, probability density functions. Become familiar with commonly encountered random variables, in particular the Gaussian random variable.
  5. Introduce the notions of expectations and moments, including means and variances. Calculate moments of common random variables. Characterize the distributions of functions of random variables.
  6. Explore the properties of multiple random variables using joint probability mass functions and joint probability density functions. Understand correlation, covariance and the correlation coefficient. Discuss how these quantities relate to the independence of random variables.
  7. Gain the ability to compute the sample mean and standard deviation of a random variable from a series of independent observations. Estimate the cumulative distribution function from a collection of independent observations. Study the law of large numbers and the central limit theorem, and illustrate how these two theorems can be employed to model random phenomena.
  8. Explain the concept of confidence intervals associated with sample means. Calculate confidence intervals and use this statistical tool to interpret engineering data.
  9. Engage the student in active learning through problem solving and real-world examples. Encourage the student to become an independent learner and increase his/her awareness of available resources.