Probability Theory and Distributions

  1. Common Discrete Distributions
    1. Bernoulli Distribution
      1. Definition and Context
        1. Single Trial Experiments
          1. Success-Failure Outcomes
          2. Parameters and Notation
            1. Probability Mass Function
              1. Mean and Variance
                1. Applications and Examples
                2. Binomial Distribution
                  1. Definition and Assumptions
                    1. Fixed Number of Trials
                      1. Independent Trials
                        1. Constant Success Probability
                        2. Parameters and Notation
                          1. Probability Mass Function
                            1. Binomial Coefficient Interpretation
                              1. Computational Aspects
                              2. Mean and Variance
                                1. Derivation Methods
                                2. Cumulative Distribution Function
                                  1. Computational Methods
                                    1. Table Usage
                                    2. Applications
                                      1. Quality Control
                                        1. Survey Sampling
                                          1. Medical Testing
                                        2. Poisson Distribution
                                          1. Definition and Context
                                            1. Rare Events
                                              1. Rate Parameter Interpretation
                                              2. Probability Mass Function
                                                1. Mean and Variance
                                                  1. Poisson Process Connection
                                                    1. Poisson Approximation to Binomial
                                                      1. Conditions for Approximation
                                                        1. Accuracy Assessment
                                                        2. Applications
                                                          1. Queueing Theory
                                                            1. Reliability Analysis
                                                              1. Epidemiology
                                                            2. Geometric Distribution
                                                              1. Definition and Context
                                                                1. Waiting Time to First Success
                                                                  1. Negative Binomial Special Case
                                                                  2. Probability Mass Function
                                                                    1. Mean and Variance
                                                                      1. Memoryless Property
                                                                        1. Statement and Proof
                                                                          1. Implications
                                                                          2. Applications
                                                                            1. Reliability Testing
                                                                              1. Customer Service
                                                                            2. Negative Binomial Distribution
                                                                              1. Definition and Context
                                                                                1. Waiting Time to r-th Success
                                                                                  1. Generalization of Geometric
                                                                                  2. Parameters and Notation
                                                                                    1. Probability Mass Function
                                                                                      1. Mean and Variance
                                                                                        1. Relationship to Other Distributions
                                                                                        2. Hypergeometric Distribution
                                                                                          1. Definition and Context
                                                                                            1. Sampling Without Replacement
                                                                                              1. Finite Population
                                                                                              2. Parameters and Notation
                                                                                                1. Probability Mass Function
                                                                                                  1. Mean and Variance
                                                                                                    1. Comparison with Binomial Distribution
                                                                                                      1. Applications
                                                                                                        1. Quality Control
                                                                                                          1. Survey Sampling
                                                                                                        2. Discrete Uniform Distribution
                                                                                                          1. Definition and Context
                                                                                                            1. Probability Mass Function
                                                                                                              1. Mean and Variance

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                                                                                                            4. Expectation and Moments

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                                                                                                            6. Common Continuous Distributions