The Poisson Process

A Poisson process with rate $\lambda$ counts arrivals over time. $N(t)$ is the number of arrivals in $[0, t]$, and it follows two rules: counts in disjoint windows are independent, and the count in any window of length $t$ is Poisson with mean $\lambda t$:

You can build it from coin flips. Cut $[0, t]$ into $n$ tiny slots and let each slot hold an arrival with probability $\lambda t/n$, independently. The count is $\text{Binomial}(n, \lambda t/n)$, and as $n \to \infty$ it becomes $\text{Poisson}(\lambda t)$, the limit from 3.0.6. The Poisson process is that same coin-flip grid run in continuous time.

Worked example: market orders hit a book at $\lambda = 4$ per minute. In one minute, $P(N = 4) = e^{-4}4^4/4! \approx 0.195$. The counts 3 and 4 tie as the most likely values, and each still happens less than a fifth of the time. In a 30-second window the mean is $2$, and $P(\text{no orders}) = e^{-2} \approx 0.135$. Mean and variance are both $\lambda t$, so over an hour you expect 240 orders, give or take about $\sqrt{240} \approx 15.5$.