UsefulLinks
Statistics
Stochastic Processes
1. Foundations of Probability Theory
2. Introduction to Stochastic Processes
3. Discrete-Time Markov Chains
4. Poisson Processes
5. Continuous-Time Markov Chains
6. Renewal Theory
7. Martingales
8. Brownian Motion
9. Stochastic Calculus
10. Stationary Processes
11. Applications in Queueing Theory
12. Applications in Finance
13. Applications in Biology and Population Dynamics
14. Applications in Physics and Engineering
4.
Poisson Processes
4.1.
Definition and Basic Properties
4.1.1.
Counting Process Definition
4.1.2.
Poisson Process Axioms
4.1.3.
Rate Parameter
4.1.4.
Stationary Increments
4.1.5.
Independent Increments
4.2.
Characterizations
4.2.1.
Exponential Inter-arrival Times
4.2.2.
Poisson Distributed Counts
4.2.3.
Uniform Order Statistics
4.3.
Inter-arrival and Waiting Times
4.3.1.
Exponential Distribution
4.3.1.1.
Memoryless Property
4.3.1.2.
Rate Parameter
4.3.2.
Waiting Time Distribution
4.3.2.1.
Gamma Distribution
4.3.2.2.
Sum of Exponentials
4.3.3.
Residual Life and Age
4.4.
Properties and Operations
4.4.1.
Superposition of Poisson Processes
4.4.1.1.
Independent Processes
4.4.1.2.
Rate Addition
4.4.2.
Thinning of Poisson Processes
4.4.2.1.
Random Selection
4.4.2.2.
Bernoulli Thinning
4.4.3.
Conditional Properties
4.4.3.1.
Given Number of Events
4.4.3.2.
Uniform Distribution Property
4.5.
Generalizations
4.5.1.
Compound Poisson Processes
4.5.1.1.
Definition and Properties
4.5.1.2.
Jump Sizes
4.5.1.3.
Lévy Processes
4.5.2.
Non-Homogeneous Poisson Processes
4.5.2.1.
Time-Dependent Rates
4.5.2.2.
Intensity Functions
4.5.2.3.
Thinning Construction
4.5.3.
Spatial Poisson Processes
4.5.3.1.
Poisson Point Processes
4.5.3.2.
Intensity Measures
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3. Discrete-Time Markov Chains
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5. Continuous-Time Markov Chains