# 3.1 Overview

## 3.1.1 Measurable outcomes

The objectives of this module are to introduce you, the student, to basic numerical methods used in probabilistic analysis and optimization, and to show you how these techniques are useful in an engineering design setting.

Specifically, students successfully completing this module will be able to:

• Measurable Outcome 3.1: Define random variables and how they can be used in mathematical modeling.

• Measurable Outcome 3.2: Define events and outcomes, list the axioms of probability.

• Measurable Outcome 3.3: Describe the process of Monte Carlo sampling from uniform distributions.

• Measurable Outcome 3.4: Describe how to generalize Monte Carlo sampling from uniform distributions to arbitrary univariate distributions.

• Measurable Outcome 3.5: Use Monte Carlo simulation to propagate uncertainty through an ODE or PDE model.

• Measurable Outcome 3.6: Describe what an estimator is.

• Measurable Outcome 3.7: Define the bias and variance of an estimator.

• Measurable Outcome 3.8: State unbiased estimators for mean and variance of a random variable, and for the probability of particular events.

• Measurable Outcome 3.9: Describe the typical convergence rate of Monte Carlo methods.

• Measurable Outcome 3.10: Define the standard error and sampling distribution of an estimator.

• Measurable Outcome 3.11: Give standard errors for sample estimators of mean, variance, and event probability.

• Measurable Outcome 3.12: Obtain confidence intervals for sample estimates of the mean, variance, and event probability.

• Measurable Outcome 3.13: Describe how to apply design of experiments methods, including parameter study, one-at-a-time, Latin hypercube sampling, and orthogonal arrays.

• Measurable Outcome 3.14: Describe the Response Surface Method.

• Measurable Outcome 3.15: Describe the construction of a response surface through Taylor series, design of experiments with least-squares regression, and random sampling with least-squares regression.

• Measurable Outcome 3.16: Describe the R2-metric, its use in measuring the quality of a response surface, and its potential problems.

• Measurable Outcome 3.17: Describe the steepest descent, conjugate gradient, and the Newton method for optimization of multivariate functions, and apply these optimization techniques to simple unconstrained design problems.

• Measurable Outcome 3.18: Describe methods to estimate gradients.

• Measurable Outcome 3.19: Use finite difference approximations to estimate gradients.

• Measurable Outcome 3.20: Interpret sensitivity information and explain its relevance to aerospace design examples.