Poisson thinning

Independently retaining a point at time $s$ with probability $p(s)$ gives a Poisson count of mean $\lambda \int p(s)ds$.

Mark each member of an independent Poisson infection cohort by its symptom-onset interval, using independent delays. Poisson thinning gives independent onset-bin counts, and sums over independent cohorts remain independent Poisson counts. Fixed cohort totals instead induce multinomial dependence. The marking assumptions are essential.

Conditioned on a total of $n$ Poisson points, the number falling in a region of relative intensity $p$ is $\mathrm{Binomial}(n,p)$.

With empty initial state, Poisson arrivals of rate $\lambda$ and exponential lifetimes of rate $\mu$ produce a Poisson occupancy of mean $\lambda (1-e^{-\mu t})/\mu$ at time $t$.