Poisson Processes

Random arrivals at a steady rate: Poisson counts, exponential gaps.

The idea

The Poisson process is the standard model for events that occur one at a time, at random moments, at a steady average rate: calls reaching a help line, buses passing a stop, atoms of a radioactive sample decaying.

To construct it, divide time into a great many tiny slices. Each slice is short enough to hold at most one arrival, each holds one with the same small probability, and the slices are independent of one another, so no arrival is ever scheduled. The number of arrivals in a window is then the number of successes in many independent, rare trials, and as the slicing refines, that count converges to a Poisson distribution.

A single parameter fixes the process: the rate $\lambda$, the average number of arrivals per unit time. The count $N(t)$ of arrivals in any window of length $t$ is Poisson with mean $\lambda t$:

$\mathbb{P}(N(t) = k) = e^{-\lambda t}\,\frac{(\lambda t)^{k}}{k!}.$

The rate and the window length enter only through the product $\lambda t$, so doubling the rate and halving the window change no probability.

Ways to work on it

• Walkthrough. Counts over a window, gaps between arrivals, and memorylessness.
• Practice. Exact count probabilities over a window.
• Hardest. Merge independent arrival streams and randomly split one.

Not sure where to start? Take the ten-question placement test.