Probability Theory

1. Probability Models
   1. Markov Chains
      1. Definition and Basic Concepts
         1. Stochastic Processes
         2. Memoryless Property (Markov Property)
      2. Transition Matrix
         1. Definition and Examples
         2. Properties
            1. Row Stochastic
            2. Irreducibility
            3. Periodicity
         3. Calculation of Future State Probabilities
      3. Classification of States
         1. Transient States
         2. Recurrent States
         3. Absorbing States
         4. Classification Theorems
      4. Steady-state Distribution
         1. Existence and Uniqueness
         2. Calculation Methods
            1. Iterative Methods
            2. Solving Linear Equations
      5. Applications
         1. PageRank Algorithm
         2. Weather Prediction Models
         3. Queueing Systems
   2. Poisson Processes
      1. Definition and Characteristics
         1. Memoryless Arrivals
         2. Homogeneous vs. Non-Homogeneous Processes
      2. Interarrival and Waiting Times
         1. Relationship to the Exponential Distribution
      3. Properties of Poisson Distribution
         1. Mean and Variance
         2. Cumulative Distribution Function
      4. Applications
         1. Modeling Telephone Call Arrivals
         2. Traffic Flow Analysis
         3. Queue Management
   3. Birth and Death Processes
      1. Definition and Structure
         1. Continuous-time Discrete-space Markov Process
      2. Transition Rates
         1. Birth Rates (λ)
         2. Death Rates (μ)
      3. Kolmogorov Forward and Backward Equations
      4. Steady-state Solutions
         1. Non-Singular Matrices for Solutions
         2. Existence Conditions
      5. Applications
         1. Population Dynamics
         2. Inventory Management
         3. Epidemiological Models
   4. Renewal Processes
      1. Definition and Conceptualization
         1. Regenerative Properties
      2. Renewal Function and Renewal Measure
         1. Direct Renewal Theorems
         2. Elementary Renewal Theorem
      3. Key Theorems
         1. Blackwell’s Theorem
         2. Smith's Renewal Theorem
      4. Applications
         1. Replacement Theory
         2. Maintenance and Reliability Analysis
         3. Insurance Risk Modelling
   5. Applications of Probability Models
      1. Inventory Systems
         1. Single Period (Newsvendor Model)
         2. Continuous Review (Q-system)
         3. Periodic Review (P-system)
      2. Queueing Theory
         1. Kendall’s Notation
         2. M/M/1 Queue System
         3. M/G/1 and G/M/1 Queues
         4. Priority Queue Systems
      3. Reliability Engineering
         1. System Reliability Modeling
         2. Redundancy Allocation
      4. Decision Theory
         1. Decision under Uncertainty
         2. Expected Utility Theory
         3. Dynamic Programming
      5. System Performance Analysis
         1. Traffic Load Balancing
         2. Network Simulation Models