Willie Wheeler's personal blog. Mostly tech.
These are some brief notes on Poisson processes, along with related processes and distributions.
Note that T and N are essentially inverses since we can recover one from the other.
The Poisson process is basically a continuous-time version of the Bernoulli trials process. Think of each Bernoulli trial as a discrete time step, and each success as an arrival. Each of the three characterizations above remains available.
|X (inter-arrival)||i.i.d., Exponential||Geometric|
|T (arrival)||Gamma||Negative binomial|
The Poisson process is wholly determined by the inter-arrival series, which is in turn under the control of a single rate parameter r.