ECE 313
Section X (Tue/Thu)

Course Outline (Tentative)

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Course Outline

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Homework Problems and Solutions

Student Projects




I. Probability Theory Basics

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

II. Conditional Probability, Independence of Events, and Bernoulli Trials

  1. Definitions of conditional probability, Bayes rule
  2. Theorem of total probability and Bayes Formula
  3. Independent events and associated rules
  4. Independence vs. Mutual exclusivity
    Mini Project 1: Analysis of alarms data from ICU patient monitoring systems
  5. Bernoulli Trials
  6. Reliability evaluation applications:
    • Series systems
    • Parallel redundancy
    • Series-parallel system evaluation
    • Non-series-parallel systems
    • Triple Modular Redundant (TMR) system with voter

III. Random Variables (Discrete)

  1. Introduction to random variables and associated event space
  2. Cumulative distribution function (CDF)
  3. Probability mass function
  4. Important discrete random variables and their distributions:
    • Bernoulli and Binomial
    • Geometric and modified geometric
    • Poisson

IV. Random Variables (Continuous)

  1. Probability density function (PDF)
  2. Important continuous random variables and their distributions:
    • Uniform
    • Gaussian (Normal)
    • Exponential
Mini Project 2

V. Exponential Distribution

  1. Memory-less property
  2. Relationship to Poisson distribution and examples
  3. Phase-type exponential distributions:
    • Hypo-exponentials
    • K-stage Erlang
    • Gamma
    • Hyper-exponential
  4. Applications to reliability evaluation

VI. Expectations:

  1. Moments: Mean and variance
  2. Mean and variance of important random variables
  3. Conditional expectation
  4. Expectation of function of random variables
  5. Covariance and correlation
  6. Reliability evaluation applications:
    • Mean time to failure
    • Failure rates
    • Hazard functions
    • Failure Data Analysis
Final Project: Analysis of Performance and Reliability of Computer Systems

VII. Joint and Conditional Density Functions

  1. Joint CFDs and PDFs
  2. Jointly Gaussian random variables
  3. Functions of many random variables
  4. Independent random variables

VIII. Binary Hypothesis Testing

IX. Inequalities and Limit Theorems

  • Markov Inequality
  • Chebyshev’s inequality
  • Law of large numbers
  • The Central Limit Theorem