Probability Theory

1. Expectation and Variance
   1. Expected Value (Mean)
      1. Definition of Expected Value
      2. Calculation of Expected Value for Discrete Random Variables
      3. Calculation of Expected Value for Continuous Random Variables
      4. Properties of Expected Value
         1. Linearity of Expectation
         2. Expectation of a Constant
      5. Applications of Expected Value
         1. Expected Return in Finance
         2. Expected Utility in Economics
   2. Variance and Standard Deviation
      1. Definition of Variance
      2. Calculation of Variance for Discrete Random Variables
      3. Calculation of Variance for Continuous Random Variables
      4. Properties of Variance
         1. Variance of a Sum of Independent Variables
         2. Variance Scaling Property
      5. Standard Deviation as a Measure of Dispersion
      6. Applications of Variance and Standard Deviation
         1. Risk Assessment in Financial Markets
         2. Quality Control in Manufacturing
   3. Properties of Expectation
      1. Linearity of Expectation: E[aX + bY + c] = aE[X] + bE[Y] + c
      2. Expectations of Sums and Differences
      3. Expectation of Products for Independent Variables
      4. Expected Value of a Function of a Random Variable
   4. Covariance
      1. Definition of Covariance
      2. Calculation of Covariance
      3. Properties of Covariance
         1. Covariance of Sums
         2. Relation to Variance
      4. Interpretation of Positive, Negative, and Zero Covariance
   5. Correlation
      1. Definition of Correlation Coefficient
      2. Calculation of Correlation
      3. Properties of Correlation
         1. Bounds of Correlation Coefficient
         2. Interpretation of Correlation Values
      4. Relationship between Correlation and Causation
   6. Law of Large Numbers
      1. Weak Law of Large Numbers
         1. Statement and Proof Outline
         2. Implications for Sample Averages
      2. Strong Law of Large Numbers
         1. Statement and Key Differences from Weak Law
         2. Applications in Convergence Theorems
      3. Importance in Statistics and Real-world Applications
   7. Applications in Expected Value and Variance
      1. Insurance Premiums and Actuarial Science
      2. Portfolio Diversification in Finance
      3. Game Theory and Decision Science
   8. Advanced Topics in Expectation and Variance
      1. Conditional Expectation
         1. Definition and Notation
         2. Applications in Probability and Statistics
      2. Moment Generating Functions and Relation to Moments
      3. Chebyshev's Inequality
         1. Statement and Applications
         2. Relationship to Probability Bounds and Variance