Unit 3: Probabilistic Methods and Optimization

3.1 Overview

• 3.1.1 Measurable Outcomes
• 3.1.2 Pre-requisite Material

3.2 Review of Probability and Statistics

• 3.2.1 Random Variables
• 3.2.2 Outcomes and Events
• 3.2.3 Distributions
• 3.2.4 Expectations
• 3.2.5 The Central Limit Theorem

3.3 Monte Carlo Methods

• 3.3.1 Introduction
• 3.3.2 Monte Carlo Analysis
• 3.3.3 Monte Carlo Example
• 3.3.4 Inversion Method for Sampling

3.4 Error Estimates for the Monte Carlo Method

• 3.4.1 The Error in Estimating the Mean
• 3.4.2 The Error in Estimating Probabilities
• 3.4.3 The Error in Estimating the Variance
• 3.4.4 Bootstrapping

3.5 Variance Reduction Techniques for the Monte Carlo Method

• 3.5.1 Monte Carlo Error as a Random Variable
• 3.5.2 Importance Sampling

3.6 Introduction to Design Optimization

• 3.6.1 Design Optimization
• 3.6.2 Gradient Based Optimization
• 3.6.3 Unconstrained Gradient-Based Optimization Methods
• 3.6.4 Finite Difference Methods
• 3.6.5 The 1d Search