Willie Wheeler's personal blog. Mostly tech.

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

- As a sequence
*X*of_{i}*inter-arrival times*, indexed by arrival*i*. - As a sequence
*T*of_{i}*arrival times*, indexed by arrival*i*. - As a random process
*N*—a_{t}*counting process*—indexed by time*t*.

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.

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*.