Probability Theory and Distributions

  1. Common Continuous Distributions
    1. Uniform Distribution
      1. Definition and Parameters
        1. Probability Density Function
          1. Cumulative Distribution Function
            1. Mean and Variance
              1. Applications
                1. Random Number Generation
                  1. Modeling Uncertainty
                2. Normal Distribution
                  1. Definition and Importance
                    1. Central Role in Statistics
                      1. Natural Occurrence
                      2. Parameters
                        1. Mean Parameter
                          1. Variance Parameter
                          2. Probability Density Function
                            1. Bell Curve Shape
                              1. Symmetry Properties
                              2. Standard Normal Distribution
                                1. Standardization Process
                                  1. Z-tables and Usage
                                    1. Probability Calculations
                                    2. Properties of Normal Distribution
                                      1. Linear Combinations
                                        1. Reproductive Property
                                          1. Tail Behavior
                                          2. Normal Approximations
                                            1. To Binomial Distribution
                                              1. To Poisson Distribution
                                                1. Continuity Correction
                                                2. Applications
                                                  1. Measurement Errors
                                                    1. Natural Phenomena
                                                      1. Quality Control
                                                    2. Exponential Distribution
                                                      1. Definition and Context
                                                        1. Continuous Analog of Geometric
                                                          1. Waiting Times
                                                          2. Parameter and Notation
                                                            1. Probability Density Function
                                                              1. Cumulative Distribution Function
                                                                1. Mean and Variance
                                                                  1. Memoryless Property
                                                                    1. Statement and Proof
                                                                      1. Practical Implications
                                                                      2. Relationship to Poisson Process
                                                                        1. Applications
                                                                          1. Reliability Engineering
                                                                            1. Queueing Systems
                                                                              1. Survival Analysis
                                                                            2. Gamma Distribution
                                                                              1. Definition and Parameters
                                                                                1. Shape Parameter
                                                                                  1. Rate Parameter
                                                                                    1. Scale Parameter Formulation
                                                                                    2. Probability Density Function
                                                                                      1. Mean and Variance
                                                                                        1. Special Cases
                                                                                          1. Exponential Distribution
                                                                                            1. Chi-squared Distribution
                                                                                            2. Gamma Function
                                                                                              1. Definition and Properties
                                                                                                1. Relationship to Factorials
                                                                                                2. Applications
                                                                                                  1. Waiting Times
                                                                                                    1. Bayesian Analysis
                                                                                                  2. Chi-squared Distribution
                                                                                                    1. Definition and Context
                                                                                                      1. Sum of Squared Standard Normals
                                                                                                        1. Degrees of Freedom
                                                                                                        2. Probability Density Function
                                                                                                          1. Mean and Variance
                                                                                                            1. Properties
                                                                                                              1. Reproductive Property
                                                                                                                1. Relationship to Gamma Distribution
                                                                                                                2. Applications
                                                                                                                  1. Goodness of Fit Tests
                                                                                                                    1. Variance Testing
                                                                                                                      1. Confidence Intervals
                                                                                                                    2. Student's t-Distribution
                                                                                                                      1. Definition and Context
                                                                                                                        1. Ratio of Normal to Chi-squared
                                                                                                                          1. Small Sample Inference
                                                                                                                          2. Degrees of Freedom Parameter
                                                                                                                            1. Probability Density Function
                                                                                                                              1. Mean and Variance
                                                                                                                                1. Relationship to Normal Distribution
                                                                                                                                  1. Convergence as df Increases
                                                                                                                                    1. Heavier Tails
                                                                                                                                    2. Applications
                                                                                                                                      1. Confidence Intervals
                                                                                                                                        1. Hypothesis Testing
                                                                                                                                          1. Small Sample Problems
                                                                                                                                        2. F-Distribution
                                                                                                                                          1. Definition and Context
                                                                                                                                            1. Ratio of Chi-squared Variables
                                                                                                                                              1. Analysis of Variance
                                                                                                                                              2. Parameters
                                                                                                                                                1. Numerator Degrees of Freedom
                                                                                                                                                  1. Denominator Degrees of Freedom
                                                                                                                                                  2. Probability Density Function
                                                                                                                                                    1. Mean and Variance
                                                                                                                                                      1. Properties
                                                                                                                                                        1. Reciprocal Property
                                                                                                                                                          1. Relationship to t-Distribution
                                                                                                                                                          2. Applications
                                                                                                                                                            1. ANOVA
                                                                                                                                                              1. Regression Analysis
                                                                                                                                                                1. Variance Comparison

                                                                                                                                                            Previous

                                                                                                                                                            5. Common Discrete Distributions

                                                                                                                                                            Go to top

                                                                                                                                                            Next

                                                                                                                                                            7. Joint Distributions and Dependence