Poisson Process

Definition 2. A counting process \(\{N(t)\}_{t\geq 0}\) is a rate-\(\lambda\) Poisson process if

1. Increments \(N(t) - N(s)\) and \(N(v) - N(u)\) are independent
2. \(P(N(t+h) - N(t) = 1) = \lambda h + o(h)\)
3. \(P(N(t+h) - N(t)\geq 2) = o(h)\)

Poisson thinning

Suppose \(N(t)\) is a Poisson process with independent events of

\[ \begin{cases} \mbox{Type 1} &\text{w.p.}\quad p \\ \mbox{Type 2} &\text{w.p.}\quad 1-p \end{cases} \]

Then, the number of type 1 events by time \(t\) (\(N_1(t)\)) and the number of type 2 events by time \(t\) (\(N_2(t)\)) are also Poisson processes of rates \(p\lambda\) and \((1-p)\lambda\) respectively.

Now suppose there are \(k\) types and the probability of an event being type \(i\) by time \(t\) is \(p_i(t)\).

Proposition. \(\displaystyle N_i(t) \sim Poisson\left(\int_{0}^{t} p_i(s) ds\right)\)

Conditioning on total events

Conditioned on \(N(t) = n\) events \(S_1,\dots S_n\), each arrival time will have a uniform distribution \(S_i\sim Unif[0, t]\).