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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.

Reference

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