Mathematics for Machine Learning and Data Science

  1. Probability Theory: Modeling Uncertainty
    1. Fundamentals of Probability
      1. Sample Spaces and Events
        1. Sample Space Definition
          1. Events and Event Space
            1. Simple and Compound Events
            2. Set Operations on Events
              1. Union of Events
                1. Intersection of Events
                  1. Complement of Events
                    1. Mutually Exclusive Events
                    2. Axioms of Probability
                      1. Non-negativity Axiom
                        1. Normalization Axiom
                          1. Additivity Axiom
                          2. Properties of Probability
                            1. Probability of Complement
                              1. Addition Rule
                                1. Inclusion-Exclusion Principle
                                2. Counting Techniques
                                  1. Multiplication Principle
                                    1. Permutations
                                      1. Combinations
                                        1. Arrangements with Repetition
                                      2. Conditional Probability and Independence
                                        1. Conditional Probability
                                          1. Definition and Interpretation
                                            1. Computing Conditional Probabilities
                                            2. Multiplication Rule
                                              1. Law of Total Probability
                                                1. Bayes' Theorem
                                                  1. Statement and Proof
                                                    1. Applications and Examples
                                                      1. Bayesian Updating
                                                      2. Statistical Independence
                                                        1. Definition for Two Events
                                                          1. Independence of Multiple Events
                                                            1. Pairwise vs Mutual Independence
                                                              1. Properties of Independent Events
                                                            2. Random Variables
                                                              1. Definition of Random Variables
                                                                1. Types of Random Variables
                                                                  1. Discrete Random Variables
                                                                    1. Continuous Random Variables
                                                                      1. Mixed Random Variables
                                                                      2. Probability Mass Functions
                                                                        1. Definition and Properties
                                                                          1. Examples and Applications
                                                                          2. Probability Density Functions
                                                                            1. Definition and Properties
                                                                              1. Relationship to Probability
                                                                              2. Cumulative Distribution Functions
                                                                                1. Definition for Discrete Variables
                                                                                  1. Definition for Continuous Variables
                                                                                    1. Properties of CDFs
                                                                                      1. Relationship between PDF and CDF
                                                                                      2. Quantile Functions
                                                                                        1. Definition and Properties
                                                                                          1. Percentiles and Quartiles
                                                                                        2. Expectation and Moments
                                                                                          1. Expected Value
                                                                                            1. Definition for Discrete Variables
                                                                                              1. Definition for Continuous Variables
                                                                                                1. Properties of Expectation
                                                                                                  1. Linearity of Expectation
                                                                                                  2. Functions of Random Variables
                                                                                                    1. Expected Value of Functions
                                                                                                      1. Law of the Unconscious Statistician
                                                                                                      2. Variance and Standard Deviation
                                                                                                        1. Definition and Computation
                                                                                                          1. Properties of Variance
                                                                                                            1. Computational Formulas
                                                                                                            2. Moments of Random Variables
                                                                                                              1. Raw Moments
                                                                                                                1. Central Moments
                                                                                                                  1. Moment Generating Functions
                                                                                                                    1. Characteristic Functions
                                                                                                                    2. Skewness and Kurtosis
                                                                                                                      1. Measures of Asymmetry
                                                                                                                        1. Measures of Tail Behavior
                                                                                                                      2. Common Probability Distributions
                                                                                                                        1. Discrete Distributions
                                                                                                                          1. Bernoulli Distribution
                                                                                                                            1. Definition and Parameters
                                                                                                                              1. Mean and Variance
                                                                                                                              2. Binomial Distribution
                                                                                                                                1. Definition and Parameters
                                                                                                                                  1. Mean and Variance
                                                                                                                                  2. Geometric Distribution
                                                                                                                                    1. Definition and Parameters
                                                                                                                                      1. Memoryless Property
                                                                                                                                      2. Negative Binomial Distribution
                                                                                                                                        1. Poisson Distribution
                                                                                                                                          1. Definition and Parameters
                                                                                                                                            1. Poisson Process
                                                                                                                                            2. Hypergeometric Distribution
                                                                                                                                              1. Discrete Uniform Distribution
                                                                                                                                              2. Continuous Distributions
                                                                                                                                                1. Uniform Distribution
                                                                                                                                                  1. Definition and Parameters
                                                                                                                                                    1. Mean and Variance
                                                                                                                                                    2. Normal Distribution
                                                                                                                                                      1. Definition and Parameters
                                                                                                                                                        1. Standard Normal Distribution
                                                                                                                                                          1. Properties and Applications
                                                                                                                                                            1. Central Limit Theorem Connection
                                                                                                                                                            2. Exponential Distribution
                                                                                                                                                              1. Definition and Parameters
                                                                                                                                                                1. Memoryless Property
                                                                                                                                                                  1. Relationship to Poisson Process
                                                                                                                                                                  2. Gamma Distribution
                                                                                                                                                                    1. Definition and Parameters
                                                                                                                                                                      1. Special Cases
                                                                                                                                                                      2. Beta Distribution
                                                                                                                                                                        1. Definition and Parameters
                                                                                                                                                                          1. Applications in Bayesian Statistics
                                                                                                                                                                          2. Chi-Square Distribution
                                                                                                                                                                            1. Student's t-Distribution
                                                                                                                                                                              1. F-Distribution
                                                                                                                                                                                1. Log-Normal Distribution
                                                                                                                                                                              2. Multiple Random Variables
                                                                                                                                                                                1. Joint Probability Distributions
                                                                                                                                                                                  1. Joint PMF for Discrete Variables
                                                                                                                                                                                    1. Joint PDF for Continuous Variables
                                                                                                                                                                                      1. Joint CDF
                                                                                                                                                                                      2. Marginal Distributions
                                                                                                                                                                                        1. Computing Marginal PMF
                                                                                                                                                                                          1. Computing Marginal PDF
                                                                                                                                                                                            1. Relationship to Joint Distributions
                                                                                                                                                                                            2. Conditional Distributions
                                                                                                                                                                                              1. Conditional PMF
                                                                                                                                                                                                1. Conditional PDF
                                                                                                                                                                                                  1. Properties of Conditional Distributions
                                                                                                                                                                                                  2. Independence of Random Variables
                                                                                                                                                                                                    1. Definition and Criteria
                                                                                                                                                                                                      1. Independence vs Uncorrelatedness
                                                                                                                                                                                                        1. Properties of Independent Variables
                                                                                                                                                                                                        2. Covariance and Correlation
                                                                                                                                                                                                          1. Covariance Definition
                                                                                                                                                                                                            1. Properties of Covariance
                                                                                                                                                                                                              1. Correlation Coefficient
                                                                                                                                                                                                                1. Properties of Correlation
                                                                                                                                                                                                                  1. Interpretation of Correlation
                                                                                                                                                                                                                  2. Bivariate Normal Distribution
                                                                                                                                                                                                                    1. Definition and Properties
                                                                                                                                                                                                                      1. Marginal and Conditional Distributions
                                                                                                                                                                                                                        1. Correlation Parameter
                                                                                                                                                                                                                      2. Limit Theorems
                                                                                                                                                                                                                        1. Law of Large Numbers
                                                                                                                                                                                                                          1. Weak Law of Large Numbers
                                                                                                                                                                                                                            1. Strong Law of Large Numbers
                                                                                                                                                                                                                              1. Applications and Interpretation
                                                                                                                                                                                                                              2. Central Limit Theorem
                                                                                                                                                                                                                                1. Statement for IID Variables
                                                                                                                                                                                                                                  1. Conditions and Assumptions
                                                                                                                                                                                                                                    1. Applications and Examples
                                                                                                                                                                                                                                      1. Approximation Quality
                                                                                                                                                                                                                                      2. Convergence of Random Variables
                                                                                                                                                                                                                                        1. Convergence in Probability
                                                                                                                                                                                                                                          1. Almost Sure Convergence
                                                                                                                                                                                                                                            1. Convergence in Distribution
                                                                                                                                                                                                                                              1. Convergence in Mean Square

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                                                                                                                                                                                                                                          5. Statistics: From Data to Insights