Statistics for Data Science

  1. Fundamentals of Probability
    1. Core Concepts
      1. Experiments, Events, and Sample Spaces
        1. Random Experiments
          1. Definition and Characteristics
            1. Deterministic vs. Random Outcomes
            2. Sample Spaces
              1. Definition and Notation
                1. Finite vs. Infinite Sample Spaces
                  1. Discrete vs. Continuous Sample Spaces
                  2. Events
                    1. Simple Events
                      1. Compound Events
                        1. Complementary Events
                          1. Mutually Exclusive Events
                          2. Event Operations
                            1. Union of Events
                              1. Intersection of Events
                                1. Complement of Events
                                  1. Difference of Events
                                2. Axioms of Probability
                                  1. Non-negativity Axiom
                                    1. Probability Range [0,1]
                                      1. Interpretation of Bounds
                                      2. Normalization Axiom
                                        1. Total Probability Equals 1
                                          1. Sample Space Probability
                                          2. Additivity Axiom
                                            1. Mutually Exclusive Events
                                              1. Countable Additivity
                                              2. Consequences of Axioms
                                                1. Complement Rule
                                                  1. Impossible Event Probability
                                                    1. Probability Bounds
                                                2. Types of Probability
                                                  1. Classical (Theoretical) Probability
                                                    1. Definition and Assumptions
                                                      1. Equally Likely Outcomes
                                                        1. Finite Sample Space
                                                        2. Calculation Methods
                                                          1. Favorable Outcomes / Total Outcomes
                                                            1. Combinatorial Approaches
                                                            2. Examples and Applications
                                                              1. Dice and Coin Problems
                                                                1. Card Drawing Problems
                                                                  1. Lottery and Gaming
                                                                2. Empirical (Frequentist) Probability
                                                                  1. Relative Frequency Approach
                                                                    1. Long-Run Frequency Interpretation
                                                                      1. Law of Large Numbers Connection
                                                                      2. Calculation Methods
                                                                        1. Observed Frequency / Total Trials
                                                                          1. Confidence in Estimates
                                                                          2. Applications
                                                                            1. Historical Data Analysis
                                                                              1. Experimental Results
                                                                                1. Quality Control
                                                                              2. Subjective Probability
                                                                                1. Personal Belief and Expert Opinion
                                                                                  1. Degree of Belief Interpretation
                                                                                    1. Prior Knowledge Incorporation
                                                                                    2. Bayesian Perspective
                                                                                      1. Prior and Posterior Probabilities
                                                                                        1. Belief Updating
                                                                                        2. Applications
                                                                                          1. Risk Assessment
                                                                                            1. Decision Making Under Uncertainty
                                                                                              1. Expert Systems
                                                                                          2. Probability Rules and Operations
                                                                                            1. Addition Rule
                                                                                              1. Mutually Exclusive Events
                                                                                                1. Simple Addition Formula
                                                                                                  1. Disjoint Events
                                                                                                  2. Non-Mutually Exclusive Events
                                                                                                    1. General Addition Rule
                                                                                                      1. Inclusion-Exclusion Principle
                                                                                                      2. Multiple Events
                                                                                                        1. Three or More Events
                                                                                                          1. Venn Diagram Applications
                                                                                                        2. Multiplication Rule
                                                                                                          1. Independent Events
                                                                                                            1. Product Rule for Independence
                                                                                                              1. Joint Probability Calculation
                                                                                                              2. Dependent Events
                                                                                                                1. Conditional Probability Integration
                                                                                                                  1. Sequential Event Analysis
                                                                                                                  2. Multiple Events
                                                                                                                    1. Chain Rule Application
                                                                                                                      1. Tree Diagram Methods
                                                                                                                    2. Complement Rule
                                                                                                                      1. Basic Complement Formula
                                                                                                                        1. Applications in Problem Solving
                                                                                                                          1. "At Least One" Problems
                                                                                                                        2. Conditional Probability
                                                                                                                          1. Definition and Calculation
                                                                                                                            1. Conditional Probability Formula
                                                                                                                              1. Interpretation and Meaning
                                                                                                                                1. Restriction of Sample Space
                                                                                                                                2. Independence of Events
                                                                                                                                  1. Definition of Independence
                                                                                                                                    1. Testing for Independence
                                                                                                                                      1. Mathematical Conditions
                                                                                                                                        1. Practical Assessment
                                                                                                                                        2. Mutual Independence
                                                                                                                                          1. Pairwise vs. Mutual Independence
                                                                                                                                            1. Multiple Event Independence
                                                                                                                                          2. Bayes' Theorem
                                                                                                                                            1. Formula Derivation
                                                                                                                                              1. From Conditional Probability
                                                                                                                                                1. Partition Theorem Connection
                                                                                                                                                2. Components
                                                                                                                                                  1. Prior Probability
                                                                                                                                                    1. Likelihood
                                                                                                                                                      1. Posterior Probability
                                                                                                                                                        1. Marginal Probability
                                                                                                                                                        2. Applications
                                                                                                                                                          1. Medical Diagnosis
                                                                                                                                                            1. Spam Filtering
                                                                                                                                                              1. Machine Learning
                                                                                                                                                              2. Real-World Examples
                                                                                                                                                                1. False Positive/Negative Rates
                                                                                                                                                                  1. Base Rate Fallacy
                                                                                                                                                                    1. Diagnostic Testing
                                                                                                                                                                  2. Law of Total Probability
                                                                                                                                                                    1. Partition of Sample Space
                                                                                                                                                                      1. Formula and Application
                                                                                                                                                                        1. Connection to Bayes' Theorem
                                                                                                                                                                      2. Combinatorics
                                                                                                                                                                        1. Fundamental Counting Principle
                                                                                                                                                                          1. Multiplication Principle
                                                                                                                                                                            1. Sequential Decision Making
                                                                                                                                                                              1. Tree Diagram Applications
                                                                                                                                                                              2. Permutations
                                                                                                                                                                                1. Ordered Arrangements
                                                                                                                                                                                  1. Definition and Formula
                                                                                                                                                                                    1. n! Factorial Notation
                                                                                                                                                                                    2. Permutations of n Objects Taken r at a Time
                                                                                                                                                                                      1. nPr Formula
                                                                                                                                                                                        1. Calculation Methods
                                                                                                                                                                                        2. Permutations with Repetition
                                                                                                                                                                                          1. Identical Objects
                                                                                                                                                                                            1. Circular Permutations
                                                                                                                                                                                          2. Combinations
                                                                                                                                                                                            1. Unordered Selections
                                                                                                                                                                                              1. Definition and Formula
                                                                                                                                                                                                1. nCr Notation
                                                                                                                                                                                                2. Relationship to Permutations
                                                                                                                                                                                                  1. Division by r! Factor
                                                                                                                                                                                                    1. Order Irrelevance
                                                                                                                                                                                                    2. Combinations with Repetition
                                                                                                                                                                                                      1. Stars and Bars Method
                                                                                                                                                                                                        1. Multiset Combinations
                                                                                                                                                                                                      2. Applications in Probability
                                                                                                                                                                                                        1. Classical Probability Calculations
                                                                                                                                                                                                          1. Sampling Without Replacement
                                                                                                                                                                                                            1. Lottery and Gaming Problems

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                                                                                                                                                                                                        2. Descriptive Statistics: Summarizing Data

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                                                                                                                                                                                                        4. Probability Distributions