Poisson Distribution

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.

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