Lecture G3 (2026-10-22): Input Modeling, Part 3 (Parameter Estimation and Goodness of Fit)

In this lecture, we aim to finish our coverage of Input Modeling, where the focus of this lecture is on parameter estimation and assessing goodness of fit. We start by identifying different shape parameters (including location, scale, and rate) as quantities that need to be estimated. Before getting to their estimation, we also discuss summary statistics (sample mean and sample variance) and maximum likelihood estimation (MLE), with an example for a point estimate of the rate of an exponential. We then discuss how AIC and BIC provide modern ways of ranking models by MLE that also account for the number of parameters in that candidate model. After covering these model selection topics, we transition to a discussion of goodness of fit. That first requires a review of the fundamentals of hypothesis testing. We discuss type-I error, p-values, type-II error, effect sizes, and statistical power. We discuss the dangers of using p-values at very large sample sizes (where small p-values are not meaningful) and at very small sample sizes (where large p-values are not meaningful). We give some examples of this applied to best-of-7 sports tournaments and voting. Finally, we close by re-introducing the chi-squared goodness of fit test for larger sample sizes and the  Kolmogorov–Smirnov (K–S, higher powered) goodness-of-fit test (and a relative, Anderson-Darling, and a distribution specific alternative for testing normality, Shapiro–Wilk) for smaller sample sizes. What we do not finish in the coverage of goodness-of-fit here will be covered in the next lecture.

Interactive widgets relevant to this lecture include:

• https://tpavlic.github.io/asu-simulating-stochastic-systems/input_modeling/input_analyzer.html
• https://tpavlic.github.io/asu-simulating-stochastic-systems/input_modeling/prob_models.html
• https://tpavlic.github.io/asu-simulating-stochastic-systems/power_analysis/power_explorer.html
• https://tpavlic.github.io/asu-simulating-stochastic-systems/output_analysis/output_analyzer/output_analyzer.html

Lecture G2 (2026-10-20): Input Modeling, Part 2 (Selection of Model Structure)

In this lecture, we continue discussing the choice of input models in stochastic simulation. Here, we pivot from talking about data collection to selection of the broad family of probabilistic distributions that may be a good fit for data. We start with an example where a histogram leads us to introduce additional input models into a flow chart. The rest of the lecture is about choosing models based on physical intuition and the shape of the sampled data (e.g., the shape of histograms). We close with a discussion of probability plots – Q-Q plots and P-P plots, as are used with "fat-pencil tests" – as a good tool for justifying the choice of a family for a certain data set. The next lecture will go over the actual estimation of the parameters for the chosen families and how to quantitatively assess goodness of fit. Concept explorers relevant to content in this lecture:

• https://tpavlic.github.io/asu-simulating-stochastic-systems/input_modeling/prob_models.html
• https://tpavlic.github.io/asu-simulating-stochastic-systems/input_modeling/input_analyzer.html

Lecture G1 (2026-10-15): Input Modeling, Part 1 (Data Collection)

In this lecture, we introduce the detailed process of input modeling. Input models are probabilistic models that introduce variation in simulation models of systems. Those input models must be chosen to match statistical distributions in data. Over this unit, we cover collection of data for this process, choice of probabilistic families to fit to these data, and then optimized parameter choice within those families and evaluation of fit with goodness of fit. In this lecture, we discuss issues related to data collection.

The Input Analyzer referenced and used in this lecture can be found at:

• https://tpavlic.github.io/asu-simulating-stochastic-systems/input_modeling/input_analyzer.html

Lecture F (2026-10-01): Midterm Review

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.

Lecture E2 (2026-09-29): Random-Variate Generation

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 Watermarking
• Concept Explorer: Probability Distributions for Input Modeling and Statistics
• Concept Explorer: Input Modeling (fully functional Input Analyzer)
  
  

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.

Lecture D2 (2026-09-22): Probabilistic Models

In this lecture, we review basic probability fundamentals (measure spaces, probability measures, random variables, probability density functions, probability mass functions, cumulative distribution functions, moments, mean/expected value/center of mass, standard deviation, variance), and then we start to build a vocabulary of different probabilistic models that are used in different modeling contexts. These include uniform, triangular, normal, exponential, Erlang-k, Weibull, and Poisson variables. We will finish the discussing next time with the Bernoulli-based discrete variables and Poisson processes.

Interactive widgets referenced or related to this lecture:

• Probability Distribution Gallery
• Input Analyzer
• Pseudo-Random Number Generator Explorer