wiki:maths:poisson

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wiki:maths:poisson [2024/11/04 20:04] – Roy Proutywiki:maths:poisson [2024/11/04 20:14] (current) – Roy Prouty
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   - 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 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?   - 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+  - 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 ==== ==== 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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 +{{ :wiki:maths:lowl.png?400 |}}
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 +{{ :wiki:maths:highl.png?400 |}}