Poisson Distribution

From Qwiki

The Poisson Distribution $P_\lambda(n,T)\,$ gives the probability for observing $n$ discrete events in a time-interval of length $T$ , when the mean number of events per unit time is $λ$ , assuming that each event occurs independently of all others in the past or future.

$P_\lambda(n,T) = \frac{e^{-\lambda T} \left(\lambda T\right)^n}{n!}$

The Poisson distribution arises all the time if you work with lasers. It looks a little strange at first, with exponentials (often associated with continuous things), factorials (often associated with discrete events), and powers (often associated with complex, multiscale phenomena). It can be derived in many ways, of course.

Derivation 1: Continuous limit of a Binomial distribution
Derivation 2: Generating function approach
Derivation 3: Summation of the waiting-time distribution