Continuous Probability: Lecture Slide Notes

These Lecture Slide Notes have been used for a two-quarter graduate level sequence in probability covering discrete and continuous probability in two separate volumes. Although reasonably self-contained, they do not constitute a formal exposition on the subject; rather the intent is to provide a concise and accessible format for reference and self-study. In this regard, each slide stands alone to encapsulate a complete concept, algorithm, or theorem, using a combination of equations, graphs, diagrams, and comparison tables. The explanatory notes are placed directly below each slide in order to reinforce key concepts and give additional insights. A Table of Contents serves to organize the slides by topic and gives a complete list of slide titles and their page numbers. An index is also provided in order to link related aspects of topics and also to cross-reference key concepts, specific applications, and the abundant visual aids. This book constitutes the second volume on continuous probability; the first volume covers discrete probability. Part 2 presupposes a working knowledge of the discrete probability concepts covered in Part 1 but is otherwise self-contained. The differential probability in an interval dx is determined by a continuous probability density function (PDF) which integrates to yield the cumulative distribution function (CDF). The concepts of joint, conditional, and marginal densities, expected values, and independence are easily transitioned to the continuous domain by emulating their discrete counterparts. The transformation between continuous probability densities is given a unique representation in terms of a composite 3-dimensional plot showing the before and after probability densities as well as the coordinate transformation curve. Both the Jacobian determinant and CDF transformation methods are covered with careful consideration of the integration and differentiation procedures involved. The CDF method for RV data simulation is motivated by a 3-dimensional plot using a "sample and hold" analog to digital coordinate transformation to generate a discrete (sampled) representation of a continuous distribution. Moment generating functions, RV sums, convolution, and "order statistics", are covered in the continuous domain, again with reference to their discrete counterparts. The distinction between counting the number events and the time between their arrivals are discussed as two complementary aspects of random processes. Continuous distributions and their relationship to limiting forms of discrete distributions are illustrated with a number of transition charts as well as a comparison of common discrete and continuous distributions. The central limit theorem, bounds for unknown distributions, and approximation methods relating sums of discrete RVs to Poisson, Gaussian, and r-Erlang estimates are also discussed. The Bivariate Gaussian distribution, its ellipses of concentration, eigenvalues, eigenvectors, and its interpretation in terms of a Bayesian measurement update for the conditional mean lead directly to the Gauss-Markov Theorem; the extension to a multivariate Gaussian distribution yields a powerful tool for multiple measurement updates in a Gaussian arena.