====== পয়সন ডিস্ট্রিবিউশন ======
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.