the Poisson process
/ pwah-SON /
Picture raindrops hitting a single roof tile, or calls arriving at a help desk, or radioactive atoms decaying one by one. The events come at a steady long-run rate, but each one lands at an unpredictable instant, and where one lands tells you nothing about when the next will come. The Poisson process is the clean mathematical model of exactly this: events scattered completely at random in time, with no clustering and no avoidance, arriving at some average rate lambda per unit time.
There are three equivalent ways to pin it down, and it is striking that they agree. (1) Count the events in any window. The number in a window of length t follows a Poisson distribution with mean lambda*t, and the counts in non-overlapping windows are independent. (2) Measure the gaps between consecutive events. These interarrival times are independent and each is exponential with rate lambda, hence memoryless. (3) Zoom into a tiny interval of length h: the chance of exactly one event is about lambda*h, the chance of two or more is negligible (order h^2), and disjoint tiny intervals act independently. Any one of these three completely determines the same process — that is what makes the Poisson process special and canonical.
It matters because 'completely random in time' is the natural null model against which real arrival patterns are judged: queues, web traffic, insurance claims, neuron firings, photon detections. The key built-in assumptions are worth stating honestly: the rate lambda is constant over time (no rush hours), events arrive one at a time (no simultaneous twins), and the future is independent of the past. When real data clusters or has a daily rhythm, you reach for the non-homogeneous or compound versions instead, but the plain Poisson process is the bedrock they all build on.
A call center receives calls as a Poisson process at lambda = 3 per minute. The number arriving in a 2-minute span is Poisson with mean 3*2 = 6. The probability that NO call arrives in the next 2 minutes is e^(-6), about 0.0025 — the same answer you get from the exponential gap or from the count, because all three definitions agree.
Counts, gaps, and infinitesimal rates: three views of one process, always agreeing.
A Poisson process is NOT 'events spaced evenly' — true randomness produces clusters and long gaps; perfectly even spacing would actually be a sign that the process is not Poisson.