Probability Theory

1. Moment Generating Functions
   1. Definition and Basic Properties
      1. Definition of Moment Generating Function (MGF)
      2. Existence Conditions for MGFs
      3. MGF of a Random Variable
      4. Relationship between MGFs and Probability Distributions
      5. Logarithmic Properties of MGFs
   2. Calculating Moments Using MGFs
      1. First Derivative of MGF for Mean (First Moment)
      2. Second Derivative of MGF for Variance (Second Moment)
      3. Higher-Order Moments
      4. Cumulant Generating Functions and Relationship to Moments
   3. Examples of MGFs
      1. MGF of Common Distributions
         1. MGF of Binomial Distribution
         2. MGF of Poisson Distribution
         3. MGF of Normal Distribution
         4. MGF of Exponential Distribution
         5. MGF of Uniform Distribution
      2. Calculation of MGFs for Multivariate Distributions
         1. Joint MGFs
         2. Product of Independent MGFs
   4. Applications of MGFs
      1. Determining Distribution Type via MGF Matching
      2. Simplifying the Process of Adding Independent Random Variables
      3. Use in Statistical Theorems and Proofs
      4. Role in Central Limit Theorem
   5. Transformations and Extensions involving MGFs
      1. Cumulant Generating Function as an Extension
      2. Moment Convergence via MGFs
      3. Extensions to Characteristic Functions
      4. Survival Analysis and Hazard Functions
   6. Limitations and Considerations
      1. Non-Existence in Some Distributions
      2. Comparison with Other Generating Functions
         1. Characteristic Functions
         2. Probability Generating Functions
      3. Interpretation Challenges with Non-Uniqueness in Inversion