ECE 313 - F, Fall 2016, University of Illinois at Urbana-Champaign

## Course Outline (Tentative)

### Exams

#### I. Introduction

1. Motivation
2. Course objectives/outline
3. Probability theory, models and their uses, examples
4. Definitions: sample space, elements, events
5. Algebra of events (union, intersections, laws/axioms)
6. Probability axioms and other useful relationships
7. Basic procedure for problem solving and an example
8. Combinatorial problems
9. Introduction to measurements

#### II. Conditional Probability and Independence of Events

1. Definitions of conditional problems, multiplication rule
2. Example
3. Independent events and associated rules
4. Application to reliability evaluation:
• Series systems
• Parallel redundancy
• Example: series-parallel system evaluation
5. Theorem of total probability, Bayes' Formula
6. Examples:
• Error-prone communication channel
• Non-series-parallel system
• Application to system reliability

#### III. Bernoulli Trials

1. TMR system with voter
2. Multiple failure modes

#### IV. Random Variables (Discrete)

1. Introduction: random variables and associated event space
2. Probability mass function
3. Special discrete random variables and their distribution:
• Binomial
• Geometric
• Poisson
• Uniform
4. Application to program/algorithmic analysis
5. Performance measurements using SPEC and other benchmarks

#### V. Random Variables (Continuous)

1. Mean, median, variance models
2. Distribution function, probability density function
3. Exponential distribution
4. Application to reliability evaluation
5. Memory less property and simple Markov model
6. Other important distributions:
• Normal
• Hyper and hypo exponentials
• Weibull
7. Expectations:
• Mean, median, variance, covariance, correlation
• Expectation of function of random variables
• Mean time to failure, Failure rates, and Hazard function
• Conditional expectation
• Inequalities and limit theorems
• Fault coverage and reliability
8. More on performance and failure measurements and analysis

#### VI. Joint Distributions

1. Joint CFDs and PDFs
2. Jointly Gaussian random variables
3. Functions of many random variables
4. Law of large numbers
5. The Central Limit Theore