wiki:maths:poisson [UMBC Observatory Wiki]

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====== Poissonian Processes ======
A Poissonian Process is one governed by Poisson Statistics. 

==== Mathematical Formulation ====

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

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:

==== Examples ====
  - Given an average number of customers per day($\lambda$), what is the probability that $x$ visit tomorrow?
  - 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?
  - 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?
  - 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.

==== Approach to Normal ====

In the limit of large average rates, the Poisson Distribution becomes indistinguishable from the [[wiki:maths:normal|Normal Distribution]].

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

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