Poissonian Processes

A Poissonian Process is one governed by Poisson Statistics.

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

With the $n$-th sampling from this distribution being noted: $x_n \thicksim \mathcal{Pois}(\lambda)$

Here, $\lambda$ is the average number of occurrences in some period, i.e., the average rate. And $x$ is the number of times an occurrence occurred in a period of equal length.

$x \thicksim \mathcal{Pois}(\lambda)$ might then be:

1. Given an average number of customers per day($\lambda$), what is the probability that $x$ visit tomorrow?
2. Given an average number of cars passing under an overpass per hour ($\lambda$), what is the probability that $x$ cars pass under an overpass from 10:43-10:44?
3. Given an average number of raindrops collected in a small container per minute ($\lambda$), what is the probability that $x$ are collected in the next minute?
4. Number of photons incident on a detector per second – teehee

We don't need to know the average number of occurrences – we just need to be able to say that these phenomena are Poisson-distributed, i.e., sampled from an unknown Poisson Distribution.

In the limit of large average rates, the Poisson Distribution becomes indistinguishable from the Normal Distribution.

That is, as $\lambda$ becomes large, $\mathcal{Pois}(\lambda) \rightarrow \mathcal{Norm}(\mu=\lambda, \sigma=\sqrt{\lambda})$