Probability Theory

1. Advanced Topics in Probability
   1. Measure Theory
      1. Sigma Algebras
         1. Definition of Sigma Algebras
         2. Properties of Sigma Algebras
            1. Closure under complementation
            2. Closure under countable unions
            3. Minimal sigma-algebra containing a collection of sets
         3. Examples of Sigma Algebras
            1. Borel Sigma Algebra
            2. Product Sigma Algebra
      2. Measurable Functions
         1. Definition and Properties
         2. Simple Functions
         3. Integration of Measurable Functions
      3. Lebesgue Integration
         1. Comparison with Riemann Integration
         2. Lebesgue’s Dominated Convergence Theorem
         3. Monotone Convergence Theorem
         4. Fatou's Lemma
         5. Applications of Lebesgue Integration in Probability
   2. Stochastic Processes
      1. Basic Concepts
         1. Definition and Classification of Stochastic Processes
         2. Stationary Processes
         3. Independent Increments
      2. Brownian Motion
         1. Properties of Brownian Motion
            1. Continuous Paths
            2. Markov Property
            3. Martingale Property
         2. Standard Brownian Motion
         3. Applications in Finance (e.g., Geometric Brownian Motion)
         4. Stochastic Calculus
            1. Ito's Lemma
            2. Stochastic Differential Equations
      3. Martingales
         1. Definition and Key Properties
            1. Fair Game Concept
            2. Stopping Time
         2. Types of Martingales
            1. Submartingales
            2. Supermartingales
         3. Martingale Convergence Theorems
         4. Applications in Finance and Gambling
   3. Information Theory
      1. Entropy
         1. Definition and Interpretation of Shannon Entropy
         2. Properties of Entropy
            1. Non-negativity
            2. Concavity
            3. Additivity
         3. Conditional Entropy
         4. Joint Entropy
      2. Mutual Information
         1. Definition and Calculation
         2. Relationship with Entropy
         3. Properties of Mutual Information
         4. Applications in Communication Systems and Machine Learning
      3. Kullback-Leibler Divergence
         1. Definition and Calculation
         2. Properties of KL Divergence
            1. Non-negativity
            2. Asymmetry
         3. Information Loss
         4. Use in Statistics and Machine Learning
            1. Model Comparison
            2. Variational Inference