Probability Theory

  1. Probability Distributions
    1. Discrete Probability Distributions
      1. Binomial Distribution
        1. Definition and Random Variables
          1. Parameters: n (number of trials), p (probability of success)
            1. Probability Mass Function (PMF) and its Derivation
              1. Mean and Variance
                1. Applications (e.g., success/failure scenarios)
                2. Poisson Distribution
                  1. Definition and Characteristics
                    1. Lambda (λ) as the Average Rate of Occurrence
                      1. PMF and its Relationship to Binomial Distribution
                        1. Mean and Variance
                          1. Applications (e.g., rare events in a fixed interval)
                          2. Geometric Distribution
                            1. Definition and Properties
                              1. Parameter: p (probability of success)
                                1. PMF and Memoryless Property
                                  1. Mean and Variance
                                    1. Applications (e.g., trials until first success)
                                    2. Hypergeometric Distribution
                                      1. Definition and Differences from Binomial
                                        1. Parameters: N (population size), n (number of successes in population), k (sample size)
                                          1. PMF and Combinatorial Interpretation
                                            1. Mean and Variance
                                              1. Applications (e.g., sampling without replacement)
                                            2. Continuous Probability Distributions
                                              1. Normal Distribution
                                                1. Definition (Gaussian Distribution)
                                                  1. Parameters: μ (mean), σ^2 (variance)
                                                    1. Probability Density Function (PDF) and Properties
                                                      1. Standard Normal Distribution (Z-distribution)
                                                        1. Central Role in the Central Limit Theorem
                                                          1. Applications (e.g., measurement errors, natural phenomena)
                                                          2. Exponential Distribution
                                                            1. Definition and Memoryless Property
                                                              1. Parameter: λ (rate parameter)
                                                                1. PDF and Cumulative Distribution Function (CDF)
                                                                  1. Mean and Variance
                                                                    1. Applications (e.g., time between events in Poisson processes)
                                                                    2. Uniform Distribution
                                                                      1. Definition and Characteristics
                                                                        1. Parameters: a (minimum), b (maximum)
                                                                          1. PDF and CDF for Continuous Uniform Distribution
                                                                            1. Mean and Variance
                                                                              1. Applications (e.g., random number generation)
                                                                              2. Gamma Distribution
                                                                                1. Definition and Extension of Exponential Distribution
                                                                                  1. Parameters: α (shape), β (rate)
                                                                                    1. Relationships with Exponential and Chi-Square Distributions
                                                                                      1. PDF and Mean and Variance
                                                                                        1. Applications (e.g., waiting time models)
                                                                                        2. Beta Distribution
                                                                                          1. Definition and Bounded Nature
                                                                                            1. Parameters: α (shape), β (shape)
                                                                                              1. PDF and the Role of Beta Function
                                                                                                1. Mean and Variance
                                                                                                  1. Applications (e.g., Bayesian statistics)
                                                                                                2. Multivariate Distributions
                                                                                                  1. Joint Distributions
                                                                                                    1. Definition and Concepts
                                                                                                      1. Joint PDF and PMF
                                                                                                        1. Marginal Distributions
                                                                                                          1. Applications in Multi-dimensional Random Variables
                                                                                                          2. Marginal Probability Distribution
                                                                                                            1. Derivation from Joint Distribution
                                                                                                              1. Importance in Independence of Variables
                                                                                                                1. Techniques for Finding Marginals
                                                                                                                2. Conditional Distribution
                                                                                                                  1. Definition and Derivation from Joint and Marginal Distributions
                                                                                                                    1. Conditional PDF and CDF
                                                                                                                      1. Relevance in Statistical Modeling
                                                                                                                    2. Special Distributions
                                                                                                                      1. t-Distribution
                                                                                                                        1. Definition and Relation to Normal Distribution
                                                                                                                          1. Parameters: degrees of freedom
                                                                                                                            1. PDF and Role in Small Sample Statistics
                                                                                                                              1. Applications (e.g., confidence intervals, hypothesis testing)
                                                                                                                              2. Chi-Square Distribution
                                                                                                                                1. Definition and Connections to Normal Distributions
                                                                                                                                  1. Degrees of Freedom as a Key Parameter
                                                                                                                                    1. PDF and Testing Goodness of Fit
                                                                                                                                      1. Applications (e.g., variance estimation)
                                                                                                                                      2. F-Distribution
                                                                                                                                        1. Definition and Ratio of Chi-Square Variables
                                                                                                                                          1. Parameters: degrees of freedom for numerator and denominator
                                                                                                                                            1. PDF and Role in ANOVA
                                                                                                                                              1. Applications (e.g., comparing variances)
                                                                                                                                          6. Expectation and Variance
                                                                                                                                          First Page
                                                                                                                                          8. Central Limit Theorem (CLT)