The Poisson Process
The Poisson process is the third basic stochastic process, the first two being the Bernoulli and Uniform processes. It can be defined in many ways, and we will examine several of them. Like other stochastic processes it has many interpretations, and each one has a large collection of applications to specific models. We also compare the Uniform and Bernoulli processes with the Poisson process. We find that the Poisson process can be regarded as a continuous analog of the Bernoulli process and that the Poisson process can be obtained in the limit from the Uniform process as the length of the interval increases. We then turn things around by showing that the Uniform process can be obtained by conditioning the Poisson process!