Review Poisson Point Process

1. History

The first literature on the poisson point process dates back to 1958[1].

2. Poisson Point Process

A process $𝑁 (𝑡)$ is called inhomogenous poisson process if:

\begin{aligned} 𝑁 (0) & = 0 \\ 𝑁 (𝑡) - 𝑁 (𝑠) & \sim Poisson (\int_{𝑠}^{𝑡} 𝜆 (𝑢) d 𝑢) for \forall 0 \leq 𝑠 < 𝑡 \end{aligned}

2.1. Likelihood

Observing events at times ${𝑡_{1}, 𝑡_{2}, \dots, 𝑡_{𝑛}}$ in a time inteveral $[0, 𝑇]$ is:

𝐿 (𝜆 (𝑡)) = exp (- \int_{0}^{𝑇} 𝜆 (𝑡) d 𝑡) \prod_{𝑖 = 1}^{𝑛} 𝜆 (𝑡_{𝑖})

Bibliography

[1] L. Mandel, Fluctuations of Photon Beams and their Correlations, Proceedings of the Physical Society 72, 1037 (1958), https://doi.org/10.1088/0370-1328/72/6/312.