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| wiki:maths:poisson [2024/11/04 19:31] – Roy Prouty | wiki:maths:poisson [2024/11/04 20:14] (current) – Roy Prouty | ||
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| ==== Mathematical Formulation ==== | ==== Mathematical Formulation ==== | ||
| - | $$\mathbf{POISSON}(\mathcal{X}=x; \lambda) = \frac{\lambda^{x}\exp^{-\lambda}}{x!}$$ | + | $$\mathcal{Pois}(\mathbf{X}=x; \lambda) = \frac{\lambda^{x}e^{-\lambda}}{x!}$$ |
| - | ==== Examples ==== | + | 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($\lambda$), | ||
| + | - Given an average number of cars passing under an overpass per hour ($\lambda$), | ||
| + | - Given an average number of raindrops collected in a small container per minute ($\lambda$), | ||
| + | - 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, | ||
| ==== Approach to Normal ==== | ==== Approach to Normal ==== | ||
| + | In the limit of large average rates, the Poisson Distribution becomes indistinguishable from the [[wiki: | ||
| + | |||
| + | That is, as $\lambda$ becomes large, $\mathcal{Pois}(\lambda) \rightarrow \mathcal{Norm}(\mu=\lambda, | ||
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