This module provides an introduction to probability that will be necessary for conducting probabilistic uncertainty quantification. The first lesson introduces basic concepts in probability starting with preliminaries in Set Theory and building up the Axioms of Probability. The second lesson then introduces random variables and defines them through their probability density function and cumulative distribution function. Moments of random variables are also introduced. The third lesson introduces random vectors and random processes to begin extending toward higher dimensional problems. Again, random vectors and random processes are defined through their distribution functions and moments are defined.

In this module, we discuss the propagation of uncertainty through a general model. We begin in Lesson 1 with simple systems where uncertainty can be propagated analytically as a function of random variables. Lesson 2 then introduces the Taylor series expansion and demonstrates how we can make a Taylor series approximation for some systems, which allows us to analytically estimate the moments of a function of random variables. Lesson 3 presents the Monte Carlo method, which is the most robust and widely-used method for propagation of uncertainty and generally serves as a benchmark against which other methods are compared. Finally, we discuss the propagation of uncertainty through the construction of surrogate models in Lesson 4. In particular, we introduce Gaussian process and polynomial chaos expansions surrogates, which are the two most commonly used approaches for uncertainty quantification.

This module provides a brief introduction into some of the more advanced topics in uncertainty quantification. Each of these topics could be covered in a course of their own, so we are only able to briefly introduce the most important concepts. The module begins by staying on the topic of uncertainty propagation and discussing advanced numerical methods - namely the spectral stochastic methods - in Lesson 1. Lesson 2 introduces reliability analysis, which is concerned with estimating small failure probabilities. Both approximate and Monte Carlo methods for reliability analysis are introduced. Lesson 3 introduces global sensitivity analysis, which aims to identify which random variables make the most significant contributions to uncertainty in the output of the model. Finally, Lesson 4 provides a brief introduction to Bayesian inference with a focus on Bayesian parameter estimation.