Stochastic Processes

  1. Poisson Processes
    1. Definition and Basic Properties
      1. Counting Process Definition
        1. Poisson Process Axioms
          1. Rate Parameter
            1. Stationary Increments
              1. Independent Increments
              2. Characterizations
                1. Exponential Inter-arrival Times
                  1. Poisson Distributed Counts
                    1. Uniform Order Statistics
                    2. Inter-arrival and Waiting Times
                      1. Exponential Distribution
                        1. Memoryless Property
                          1. Rate Parameter
                          2. Waiting Time Distribution
                            1. Gamma Distribution
                              1. Sum of Exponentials
                              2. Residual Life and Age
                              3. Properties and Operations
                                1. Superposition of Poisson Processes
                                  1. Independent Processes
                                    1. Rate Addition
                                    2. Thinning of Poisson Processes
                                      1. Random Selection
                                        1. Bernoulli Thinning
                                        2. Conditional Properties
                                          1. Given Number of Events
                                            1. Uniform Distribution Property
                                          2. Generalizations
                                            1. Compound Poisson Processes
                                              1. Definition and Properties
                                                1. Jump Sizes
                                                  1. Lévy Processes
                                                  2. Non-Homogeneous Poisson Processes
                                                    1. Time-Dependent Rates
                                                      1. Intensity Functions
                                                        1. Thinning Construction
                                                        2. Spatial Poisson Processes
                                                          1. Poisson Point Processes
                                                            1. Intensity Measures

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                                                        3. Discrete-Time Markov Chains

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                                                        5. Continuous-Time Markov Chains