Stochastic Processes

A stochastic process is a collection of random variables, representing the evolution of a system over time or space in a way that incorporates inherent randomness. Unlike analyzing a single, static random event, a stochastic process models the entire sequence of outcomes, where the state of the system at any point is governed by probabilistic rules rather than being deterministic. These processes are fundamental tools for modeling and predicting dynamic phenomena where chance plays a key role, with common examples including the fluctuating price of a stock, the random walk of a particle in Brownian motion, or the number of customers waiting in a queue.

1. Foundations of Probability Theory
   1. Sample Spaces and Events
      1. Sample Space Definition
      2. Event Definition and Notation
      3. Types of Events
         1. Simple Events
         2. Compound Events
         3. Mutually Exclusive Events
         4. Exhaustive Events
         5. Complementary Events
      4. Set Operations on Events
         1. Union of Events
         2. Intersection of Events
         3. Difference of Events
   2. Probability Axioms and Properties
      1. Kolmogorov's Axioms
         1. Non-negativity Axiom
         2. Normalization Axiom
         3. Countable Additivity Axiom
      2. Properties Derived from Axioms
         1. Probability of Empty Set
         2. Probability of Complement
         3. Monotonicity Property
         4. Inclusion-Exclusion Principle
      3. Finite and Countable Sample Spaces
   3. Conditional Probability and Independence
      1. Definition of Conditional Probability
      2. Multiplication Rule
      3. Law of Total Probability
      4. Independence of Events
         1. Definition of Independence
         2. Pairwise Independence
         3. Mutual Independence
         4. Independence of Collections of Events
      5. Bayes' Theorem
         1. Statement and Derivation
         2. Applications and Examples
   4. Random Variables
      1. Definition and Basic Properties
      2. Types of Random Variables
         1. Discrete Random Variables
         2. Continuous Random Variables
         3. Mixed Random Variables
      3. Functions of Random Variables
   5. Discrete Random Variables
      1. Probability Mass Function
         1. Definition and Properties
         2. Normalization Condition
      2. Common Discrete Distributions
         1. Bernoulli Distribution
         2. Binomial Distribution
         3. Geometric Distribution
         4. Negative Binomial Distribution
         5. Poisson Distribution
         6. Hypergeometric Distribution
   6. Continuous Random Variables
      1. Probability Density Function
         1. Definition and Properties
         2. Normalization Condition
      2. Common Continuous Distributions
         1. Uniform Distribution
         2. Normal Distribution
         3. Exponential Distribution
         4. Gamma Distribution
         5. Beta Distribution
         6. Chi-Square Distribution
   7. Cumulative Distribution Function
      1. Definition and Properties
      2. Relationship to PMF and PDF
      3. Right-Continuity Property
      4. Inverse CDF and Quantiles
   8. Expectation and Moments
      1. Expected Value
         1. Definition for Discrete Variables
         2. Definition for Continuous Variables
         3. Linearity of Expectation
         4. Expectation of Functions of Random Variables
      2. Variance and Standard Deviation
         1. Definition and Properties
         2. Computational Formulas
         3. Variance of Linear Combinations
      3. Higher-Order Moments
         1. Raw Moments
         2. Central Moments
         3. Skewness and Kurtosis
      4. Moment Generating Functions
         1. Definition and Properties
         2. Uniqueness Theorem
         3. MGFs of Common Distributions
   9. Multiple Random Variables
      1. Joint Distributions
         1. Joint PMF for Discrete Variables
         2. Joint PDF for Continuous Variables
         3. Joint CDF
      2. Marginal Distributions
         1. Marginalization Process
         2. Marginal PMFs and PDFs
      3. Conditional Distributions
         1. Conditional PMFs and PDFs
         2. Conditional Expectation
      4. Independence of Random Variables
         1. Definition and Criteria
         2. Independence of Functions
      5. Covariance and Correlation
         1. Definition of Covariance
         2. Properties of Covariance
         3. Correlation Coefficient
         4. Linear Relationships
   10. Limit Theorems
       1. Convergence Concepts
          1. Convergence in Distribution
          2. Convergence in Probability
          3. Almost Sure Convergence
          4. Convergence in Mean Square
       2. Law of Large Numbers
          1. Weak Law of Large Numbers
          2. Strong Law of Large Numbers
          3. Applications and Interpretations
       3. Central Limit Theorem
          1. Classical Central Limit Theorem
          2. Lindeberg-Lévy Theorem
          3. Applications and Approximations
   11. Conditional Expectation
       1. Conditioning on Events
       2. Conditioning on Random Variables
       3. Properties of Conditional Expectation
          1. Linearity
          2. Tower Property
          3. Jensen's Inequality
       4. Law of Total Expectation
       5. Law of Total Variance