Probability Theory

1. Random Variables
   1. Introduction to Random Variables
      1. Definition and Understanding
         1. Concept of a Random Variable
         2. Mapping from Sample Space to Real Line
      2. Classification
         1. Discrete vs Continuous
         2. Characteristics and Examples
   2. Discrete Random Variables
      1. Properties of Discrete Random Variables
         1. Countable Outcomes
         2. Examples and Applications
      2. Probability Mass Function (PMF)
         1. Definition and Characteristics
         2. Calculations and Examples
         3. Properties of PMF
            1. Non-negativity
            2. Sum to One
      3. Cumulative Distribution Function (CDF)
         1. Definition and Relation to PMF
         2. Steps for Calculation
         3. Properties of CDF
            1. Non-decreasing Function
            2. Limiting Values
   3. Continuous Random Variables
      1. Characteristics of Continuous Random Variables
         1. Uncountable Outcomes
         2. Real-world Examples
      2. Probability Density Function (PDF)
         1. Definition and Visualization
         2. Relationship with the CDF
         3. Properties of PDF
            1. Non-negativity
            2. Integration Over the Range
      3. CDF for Continuous Variables
         1. Calculating Probability Using CDF
         2. Connection with PDF
         3. Properties and Graphical Representations
   4. Joint Random Variables
      1. Understanding Joint Random Variables
         1. Definition and Explanation
         2. Examples with Two or More Variables
      2. Joint Probability Distributions
         1. Joint PMF for Discrete Random Variables
         2. Joint PDF for Continuous Random Variables
      3. Independence of Random Variables
         1. Criteria and Testing for Independence
         2. Examples and Implications
   5. Transformation of Random Variables
      1. Techniques for Transformation
         1. Functions of Random Variables
         2. Deriving New Variables
      2. Applications and Examples
         1. Linear Transformations
         2. Non-linear Transformations
   6. Moments of Random Variables
      1. Definition and Importance of Moments
      2. Types of Moments
         1. Mean (First Moment)
         2. Variance (Second Moment)
         3. Higher-order Moments
      3. Calculations and Interpretations
         1. Expectation and Variance Calculations
         2. Skewness and Kurtosis
   7. Functions of Random Variables
      1. Finding Distributions of Functions
      2. Methods and Theorems
         1. Distribution Function Technique
         2. Change of Variables
   8. Convergence of Random Variables
      1. Types of Convergence
         1. Convergence in Probability
         2. Almost Sure Convergence
         3. Convergence in Distribution
      2. Limit Theorems and Applications
         1. Central Limit Theorem
         2. Weak Law of Large Numbers
   9. Random Vectors
      1. Extension to Multivariate Random Variables
      2. Definitions and Notations
      3. Joint and Marginal Distributions
         1. Multivariate Joint Distributions
         2. Conditional Distributions
   10. Conditional Expectation
       1. Understanding Conditional Expectation
       2. Calculating Conditional Expectations
       3. Applications and Theoretical Importance