Poissonian Processes

A Poissonian Process is one governed by Poisson Statistics.

$$\mathcal{POISSON}(\mathbf{X}=x; \lambda) = \frac{\lambda^{x}e^{-\lambda}}{x!}$$

With the $n$-th sampling from this distribution being noted: $\mathbf{X}_n \thicksim \mathcal{POISSON}(\mathbf{X}=x; \lambda)$