wiki:maths:poisson

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
Next revision		 Previous revision
wiki:maths:poisson [2024/11/04 19:41] – Roy Proutywiki:maths:poisson [2024/11/04 20:14] (current) – Roy Prouty
Line 6: Line 6:
 $$\mathcal{Pois}(\mathbf{X}=x; \lambda) = \frac{\lambda^{x}e^{-\lambda}}{x!}$$ $$\mathcal{Pois}(\mathbf{X}=x; \lambda) = \frac{\lambda^{x}e^{-\lambda}}{x!}$$
  
-With the $n$-th sampling from this distribution being noted: $x_n \thicksim \mathcal{Pois}(\lambda)$+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.+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.\\ \\
  
-$x_n \thicksim \mathcal{Pois}(\lambda)$ might then be:+$x \thicksim \mathcal{Pois}(\lambda)$ might then be:
  
 ==== Examples ==== ==== Examples ====
-  - Given an average number of customers per day($\lambda$), what is the probability that $x_n$ visit tomorrow? +  - 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 $x_n$ 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 $x_n$ 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 ====
  
 +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, $\mathcal{Pois}(\lambda) \rightarrow \mathcal{Norm}(\mu=\lambda, \sigma=\sqrt{\lambda})$
  
-==== Approach to Normal ====+{{ :wiki:maths:lowl.png?400 |}}
  
 +{{ :wiki:maths:highl.png?400 |}}