Probability Theory and Distributions

Probability Theory and Distributions form the mathematical foundation of statistics, providing a rigorous framework for quantifying uncertainty and modeling random phenomena. This field establishes the fundamental axioms and rules for calculating the likelihood of events, while the study of distributions—such as the Normal, Binomial, and Poisson—provides specific mathematical models that describe how probabilities are allocated across all possible outcomes of a random variable. These concepts are essential for statistical inference, enabling the analysis of data, the testing of hypotheses, and the making of predictions with a specified degree of confidence.

1. Foundational Concepts of Probability
   1. Random Experiments and Sample Spaces
      1. Definition of Random Experiments
         1. Characteristics of Random Experiments
         2. Repeatability Under Similar Conditions
         3. Unpredictable Individual Outcomes
         4. Well-Defined Set of Possible Outcomes
      2. Examples of Random Experiments
         1. Coin Tossing
         2. Die Rolling
         3. Card Drawing
         4. Measurement Experiments
      3. Sample Space
         1. Definition and Notation
         2. Finite Sample Spaces
            1. Counting Elements
            2. Enumeration Methods
         3. Countably Infinite Sample Spaces
            1. Natural Number Sequences
            2. Geometric Progressions
         4. Uncountably Infinite Sample Spaces
            1. Continuous Intervals
            2. Multi-dimensional Spaces
      4. Sample Points and Outcomes
         1. Elementary Outcomes
         2. Composite Outcomes
         3. Notation and Representation
   2. Events and Event Algebra
      1. Definition of Events
         1. Events as Subsets of Sample Space
         2. Simple Events
         3. Compound Events
      2. Special Events
         1. Null Event (Empty Set)
         2. Universal Event (Sample Space)
         3. Singleton Events
      3. Event Operations
         1. Union of Events
            1. Definition and Notation
            2. Venn Diagram Representation
            3. Properties of Union
         2. Intersection of Events
            1. Definition and Notation
            2. Venn Diagram Representation
            3. Properties of Intersection
         3. Complement of Events
            1. Definition and Notation
            2. Properties of Complement
         4. Difference of Events
            1. Definition and Notation
            2. Relationship to Intersection and Complement
         5. Symmetric Difference
            1. Definition and Properties
      4. Event Relationships
         1. Mutually Exclusive Events
            1. Definition and Examples
            2. Pairwise Disjointness
            3. Complete Disjointness
         2. Exhaustive Events
            1. Definition and Properties
            2. Partition of Sample Space
         3. Subset Relationships
            1. Event Inclusion
            2. Proper Subsets
   3. Probability Axioms and Basic Properties
      1. Kolmogorov Axioms
         1. Axiom of Non-negativity
            1. Statement and Interpretation
         2. Axiom of Normalization
            1. Probability of Sample Space
         3. Axiom of Countable Additivity
            1. Statement for Disjoint Events
            2. Extension to Infinite Collections
      2. Immediate Consequences of Axioms
         1. Probability of Empty Set
         2. Probability of Complement
         3. Finite Additivity
         4. Monotonicity Property
      3. Fundamental Probability Rules
         1. Addition Rule for Two Events
            1. General Form
            2. Special Case for Disjoint Events
         2. Inclusion-Exclusion Principle
            1. Two Events
            2. Three Events
            3. General Form for n Events
         3. Bonferroni Inequalities
            1. Upper and Lower Bounds
         4. Subadditivity Property
            1. Statement and Applications
   4. Methods of Probability Assignment
      1. Classical Probability
         1. Equally Likely Outcomes Assumption
         2. Counting Favorable Outcomes
         3. Limitations and Assumptions
      2. Combinatorial Methods
         1. Fundamental Counting Principle
         2. Permutations
            1. Without Repetition
            2. With Repetition
            3. Circular Permutations
         3. Combinations
            1. Without Repetition
            2. With Repetition
            3. Properties of Binomial Coefficients
      3. Relative Frequency Approach
         1. Empirical Probability
         2. Stabilization of Relative Frequencies
         3. Law of Large Numbers Connection
      4. Subjective Probability
         1. Personal Probability Assessment
         2. Coherence Requirements
         3. Bayesian Framework Introduction