During this lecture, we review the topics covered up to this point in the course as preparation for the upcoming midterm exam. Students are encouraged to bring their own questions to class so that we can focus on the topics that students feel like they need the most help with.
In this lecture, we review pseudo-random number generation and then introduce random-variate generation by way of inverse-transform sampling. In particular, we start with a review of the two most important properties of a pseudo-random number generator (PRNG), uniformity and independence, and discuss statistically rigorous methods for testing for these two properties. For uniformity, we focus on a Chi-square/Chi-squared test for larger numbers of samples and a Kolmogorov–Smirnov (KS) test for smaller numbers of samples. For independence, we discuss autocorrelation tests and runs test, and then we demonstrate a runs above-and-below-the-mean test. We then shift to discussing inverse-transform sampling for continuous random variates and discrete random variates and how the resulting random-variate generators might be implemented in a tool like Rockwell Automation's Arena.Concept explorers relevant to content in this lecture:Concept Explorer: Pseudo-Random Number Generators and AI WatermarkingConcept Explorer: Probability Distributions for Input Modeling and StatisticsConcept Explorer: Input Modeling (fully functional Input Analyzer)
24 Sept 2026
Lecture E1 (2026-09-04): Random-Number Generation
In this lecture, we first cover some discrete distributions (and the Poisson process) that we ran out of time for during the previous lecture. We then launch into a discussion of how to generate pseudo-random numbers distributed uniformly between 0 and 1 (which are necessary for us to easily generate random variates of any distribution). We talk about the two most important properties of a pseudo-random number generator (PRNG), uniformity and independence. We then talk about desirable properties. Some examples are given of some early PRNG's, and then we introduce the linear congruential generator (LCG) and its variants (like the Combined Linear Congruential Generator, CLCG), which represent a much more modern PRNG that has a number of good properties. We close with a discussion of tests of uniformity. We will continue this discussion and add on tests for independence during next lecture (which will primarily cover random-VARIATE generation). Students may wish to follow up by checking out the Pseudo-Random Number Generators and AI Watermarking concept explorer, which covers LCG's, CLCG's, and the more modern MRG approach to PRNG. It also covers methods for AI watermarking of stochastic outputs, like the ones that are now used in large language models like Gemini and Claude.
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