Willie Wheeler's personal blog. Mostly tech.

Poisson Process and Related Distributions

These are some brief notes on Poisson processes, along with related processes and distributions.

Different ways to characterize the Poisson process

  • As a sequence Xi of inter-arrival times, indexed by arrival i.
  • As a sequence Ti of arrival times, indexed by arrival i.
  • As a random process Nt—a counting process—indexed by time t.

Note that T and N are essentially inverses since we can recover one from the other.

Poisson vs Bernoulli trials process

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.

Probability distributions by process and series

Series Poisson Bernoulli
X (inter-arrival) i.i.d., Exponential Geometric
T (arrival) Gamma Negative binomial
N (count) Poisson 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.