Today I Learned

Some of the things I've learned every day since Oct 10, 2016

19: Poisson Distribution

The distribution of a discrete random variable $X$ is considered a Poisson Distribution if, given an event $x$ an given interval $N$ (which can be thought of a number of chances $x$ is given to occur), the probability of $x$ occurring exactly $k$ times on that size interval is given by

$P(k \textrm{ occurrences}) = \frac{\lambda ^k e^{- \lambda}}{k!}$,

where $\lambda$ is the known average of occurrences given an identical interval.

[Restrictions: events must occur independently, and no more than one instance of an event can happen at a single point in the interval.]

A couple nice properties of Poisson-distributed random variables are that the expected value and variance are both equal to $\lambda$.