superposition and thinning of Poisson processes

Two operations make Poisson processes wonderfully friendly. Superposition is pooling: merge the arrivals of several independent Poisson streams into one combined stream — for example, calls to a switchboard coming from three independent regions, all logged together. Thinning is the reverse, a filter: keep each arrival of one Poisson stream at random (toss a coin for each event, keep it with some probability) and discard the rest. The headline result is that both operations keep you inside the Poisson family.

Superposition: if you merge independent Poisson processes with rates lambda1, lambda2, ..., the combined process is again Poisson with rate lambda1 + lambda2 + ... The rates simply add, which mirrors the fact that independent Poisson counts add. Thinning: take a Poisson process of rate lambda and independently keep each event with probability p (discard with probability 1 - p). The kept events form a Poisson process of rate lambda*p, the discarded events form a Poisson process of rate lambda*(1 - p) — and, strikingly, these two thinned processes are INDEPENDENT of each other, even though they came from the same original. More generally, classifying each event into one of several types with fixed probabilities splits one Poisson process into independent Poisson processes, one per type.

These rules turn hard problems into easy bookkeeping. Web traffic from many independent sources is Poisson with the summed rate; if 2 percent of arriving customers are VIPs, the VIP arrivals are their own Poisson process at rate 0.02*lambda, independent of the rest. The one assumption to honor is independence: superposition needs the streams to be independent, and thinning needs each keep/discard decision to be independent of the timing. Dependence (e.g. a burst that makes both a customer arrive AND be a VIP) breaks the clean Poisson conclusion.