The Poisson distribution describes the probability of \( n \) events occurring in a given interval \( A \) when \( \nu \) are to be expected. The Poisson distribution is discrete, as the number of events that are observed can only be an integer. The expectation value \( \nu \) on the other hand can be contiguous. As the standard deviation is a function of the expected number of events \( \nu \) and \(\nu \) is the mean, the Poisson distribution is has only one parameter, namely \(\nu \). The standard deviation in terms of \( \nu \) is defined as: \(\sigma = \sqrt{\nu} \).

The Poisson distribution is often used in nuclear and particle physics. A Poisson distribution most common use is to give the uncertainty for histogram bins as the number of events in a histogram bin is exactly the answer to the question of how many events are observed within the bin-boundaries. The error for the histogram bins can therefore be easily calculated.

The Poisson distribution is defined as:

$$ f(n, \nu) = \frac{\nu^n}{n!} e^{-\nu} $$

For large \( \nu \) the poisson tends towards the Gaussian distribution following the central limit theorem.

Plots

Poisson distributions with different nu values
The Poisson distribution with different values for nu