13. The Poisson Process

Summary

The Poisson process is one of the most important random processes in probability theory. It is widely used to model random points in time and space, such as the times of radioactive emissions, the arrival times of customers at a service center, and the positions of flaws in a piece of material. Several important probability distributions arise naturally from the Poisson process—the Poisson distribution, the exponential distribution, and the gamma distribution. The process has a beautiful mathematical structure, and is used as a foundation for building a number of other, more complicated random processes.

Basic Topics

1. Introduction
2. The Exponential Distribution
3. The Gamma Distribution
4. The Poisson Distribution

Variations

5. Thinning and Superpositon
6. Non-homogeneous Poisson Processes
7. Compound Poisson Processes
8. Poisson Processes on General Spaces

Apps

• Gamma Experiment
• Poisson Experiment
• Two-Type Poisson Experiment
• Two-Dimensional Poisson Process

Sources and Resources

• Introduction to Probability Models. Sheldon Ross
• Stochastic Processes. Sheldon Ross
• A First Course in Stochastic Processes. Samuel Karlin and Howard Taylor
• Introduction to Stochastic Processes. Erhan Çinlar
• Poisson Processes. JFC Kingman
• Adventures in Stochastic Processes, Sidney I Resnick
• Probability and Random Processes. Geoffrey Grimmett and David Stritzaker