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ATI's Fundamentals of Engineering Probability
Visualization Techniques & MatLab Case Studies

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    Technical Training Short On Site Course Quote This four-day course gives a solid practical and intuitive understanding of the fundamental concepts of discrete and continuous probability. It emphasizes visual aspects by using many graphical tools such as Venn diagrams, descriptive tables, trees, and a unique 3-dimensional plot to illustrate the behavior of probability densities under coordinate transformations. Many relevant engineering applications are used to crystallize crucial probability concepts that commonly arise in aerospace CONOPS and tradeoffs. View Course Sampler

    Dr. Ralph E. Morganstern is an Adjunct Lecturer in Applied Mathematics at Santa Clara University where he teaches graduate-level sequences in Probability and Numerical Analysis. Dr. Morganstern received a Ph.D. in Physics from the State University of New York at Stony Brook. He has published papers on general relativity, astrophysics, and cosmology and served as a referee on The Physical Review and The Astrophysical Journal. Dr. Morganstern has worked in the Aerospace Industry in Silicon Valley California for over 30 years. He has applied fundamental physics concepts to formulate mathematical models and develop efficient algorithms in many engineering areas including image enhancement, atmospheric optics, data fusion, satellite tracking, communications, and SAR and FMCW radar processing.

    Contact this instructor (please mention course name in the subject line)

What You Will Learn:

  • How to compute joint, conditional, and marginal probability densities.
  • How to compute & visualize probability densities of transformed RVs.
  • How to sum dB-scaled measurements to make sequential Bayesian updates.
  • How to compute approximations/upper bounds on sums of many RVs using Gaussian and Poisson distributions.
  • How the bivariate Gaussian is totally characterized by its mean vector and the covariance matrix between its two independent RVs.
  • How the Gauss-Markov theorem yields a conditional mean estimator for vector measurements and vector states.
  • This course will de-mystify the computational aspects associated with the transformation of multivariate probability densities and give you the confidence to analyze the random variable effects that arise in engineering scenarios.

Course Outline:

  1. Probability and Counting. Visualizations via coordinate graphs, tables, trees, and Venn diagrams. Set theory concepts. DeMorgans Rules. Role of Mutually Exclusive (ME) and Collectively Exhaustive (CE) event spaces. Sample Space with equally likely outcomes. Probability computed via combinatorial analysis.
  2. Fundamentals of Probability. Axioms of probability. Classical, Frequentist, Bayesian, and ad hoc probability frameworks. Mutually exclusive versus independent events. Inclusion/Exclusion concepts and applications. Comparison of tree, tabular, Venn, and algebraic representations (Man-Hat problem). Conditional probability and its tree interpretation. Repeated independent trials. Binomials, Trinomials, Multinomials. System reliability analysis.
  3. Random Variables and Probability Distributions. Random variable probability mass functions (PMFs) and cumulative distribution functions (CDFs). Joint, marginal, and conditional distributions. Discrete RVs under a transformation of coordinates. Distributions for derived RVs. 4-sided dice sum/difference coordinates. Min & max coordinates and order statistics. Mean variance, covariance and linear transformations.
  4. Common PMFs. Pairs: {Bernoulli, Binomial} & {Geometric, Negative Binomial}. Common Characteristics: {Hyper-geometric, Poisson, Zeta(Zipf)}. Properties, relationships, plots, and trees. Statistical analysis of Bernoulli Trials. Sum of RVs, convolution. Moment generating function. Engineering examples.
  5. Transition to Continuous Probability Concepts. Continuous & mixed probability densities in 1 & 2 dimensions. Dirac delta function and Heaviside step function. Probability Density Function (PDF) and Cumulative Distribution Function (CDF) for continuous and mixed distributions. Density transformation techniques: Jacobian Method. CDF method. 3-dimensional visualizations of density transformations. Order statistics for continuous variables. DSP chip with uniform interrupts. Generating Function, RV Sums, and Convolution.
  6. Random Processes. Taxonomy of random processes. Bernoulli to Gaussian & Poisson. Sum of Bernoulli RVs to Binomial. Sum of Geometric RVs to Negative Binomial. Discrete Poisson & continuous r-Erlang relationship. Gaussian distribution & standardized variable. Normal Distribution standard table. Continuous PDFs: Uniform, Exponential, Gamma(r-Erlang), Normal, Rayleigh. Properties, relationships, plots, and examples
  7. Approximations & Bounds. Central Limit Theorem, Approximation Techniques for Binomial & Poisson Distributions. DeMoivre-Laplace approximation. Markov & Chebyshev Bounds. Law of Large Numbers.
  8. Bivariate & Multivariate Gaussian Distributions. Matrix form of bivariate Gaussian distribution. Transformation of coordinates & covariance matrix. Ellipses of concentration. Standardized look-up table for 2d Gaussians. Covariance Matrix eigenvector-eigenvalue problem. Canonical coordinates & independence. Bayesian update - conditional mean interpretation and visualization. Multivariate Gaussian. Canonical Block diagonal form. Channel & inverse-channel representations. Gauss-Markov theorem for vectorized conditional mean.
  9. MatLab Case Studies. Line of sight error analysis for satellite and ocean sensors. Effects of long-tailed duration distributions on Internet Flows. Statistical air traffic pattern generator.


    Tuition for this four day course is $1995 per person at one of our scheduled public courses. Onsite pricing is available. Please call us at 410-956-8805 or send an email to

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