Mathematical Foundations for Computing

  1. Discrete Probability
    1. Introduction to Discrete Probability
      1. Experiments, Sample Spaces, and Events
        1. Random Experiments
          1. Sample Space
            1. Events as Subsets
              1. Event Operations
              2. Probability of an Event
                1. Definition of Probability
                  1. Properties of Probability
                  2. Assigning Probabilities
                    1. Equally Likely Outcomes
                      1. Relative Frequency
                        1. Subjective Probability
                        2. Finite Probability Spaces
                          1. Uniform Probability Models
                            1. Non-uniform Models
                          2. Probability Theory
                            1. Axioms of Probability
                              1. Kolmogorov Axioms
                                1. Consequences of Axioms
                                2. Probability of Complements
                                  1. Complement Rule
                                  2. Probability of Unions of Events
                                    1. Addition Rule
                                      1. Mutually Exclusive Events
                                      2. Probability of Intersections of Events
                                        1. Multiplication Rule
                                          1. Independent Events
                                          2. Conditional Probability
                                            1. Definition and Calculation
                                              1. Multiplication Rule
                                                1. Law of Total Probability
                                                2. Independence of Events
                                                  1. Definition of Independence
                                                    1. Pairwise Independence
                                                      1. Mutual Independence
                                                        1. Testing for Independence
                                                      2. Bayes' Theorem
                                                        1. Statement and Proof
                                                          1. Prior and Posterior Probabilities
                                                            1. Applications of Bayes' Theorem
                                                              1. Medical Testing
                                                                1. Spam Filtering
                                                                2. Random Variables
                                                                  1. Definition of Random Variable
                                                                    1. Discrete Random Variables
                                                                      1. Probability Mass Function
                                                                      2. Discrete Probability Distributions
                                                                        1. Uniform Distribution
                                                                          1. Bernoulli Distribution
                                                                            1. Binomial Distribution
                                                                              1. Geometric Distribution
                                                                                1. Poisson Distribution
                                                                                2. Expected Value
                                                                                  1. Definition and Calculation
                                                                                    1. Linearity of Expectation
                                                                                      1. Expected Value of Functions
                                                                                      2. Variance and Standard Deviation
                                                                                        1. Definition of Variance
                                                                                          1. Calculation Methods
                                                                                            1. Standard Deviation
                                                                                              1. Properties of Variance

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