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--- un:poisson-distribution [2023/06/20 09:52] – asad
+++ un:poisson-distribution [2023/10/22 05:16] (current) – removed asad
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-====== Poisson distribution ======
-A discrete [[probability distribution]] that expresses the probability of a given number of events occurring in a fixed interval of time or space when the events occur independently (non-correated) and at a constant given average rate. It is named after the French mathematician Siméon Denis Poisson (1781--1840).
-If the average (mean) rate is  $\mu$ , the probability of counting  $x$  events in a single trial
- $P(x,\mu) = \frac{\mu^x}{x!} e^{-\mu}$
-where  $e$  is [[Euler's number]] and  $x$  is a non-negative integer. The function is plotted below for three different means.
-[[https://colab.research.google.com/drive/1_Qwr1tiXnFgphLiypeGJcEmkLMMOHP41?usp=sharing|{{:un:poisson.png?nolink&750|}}]]
-For example, if  $\mu=2.8$  (green curve) raindrops are falling on a tin roof on average per second, then the probability of hearing 4 raindrops in the next second  $P(4,2.8) \approx 0.15$ .
-The variance of Poisson distribution  $\sigma^2=\mu$  and, hence, the standard deviation  $\sigma=\sqrt{\mu}$ .
-The fractional uncertainty in counting  $N$  events
- $\frac{\sigma}{\mu} \approx \frac{1}{\sqrt{N}}.$
-If  $N$  photons have arrived in the detector of a telescope in  $t$  seconds, then the //counts per second//  $N/t$  has an uncertainty  $\sigma/t \approx \sqrt{N}/t$ . In order to decrease the uncertainty in measuring photons, we have to increase the number of detected photons by either increasing the size of the telescope or the exposure time.