Definition:Poisson Distribution

Definition
Let $X$ be a discrete random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.

Then $X$ has the Poisson distribution with parameter $\lambda$ (where $\lambda > 0$) if:


 * $\Img X = \set {0, 1, 2, \ldots} = \N$


 * $\map \Pr {X = k} = \dfrac 1 {k!} \lambda^k e^{-\lambda}$

Note that Poisson Distribution Gives Rise to Probability Mass Function satisfying $\map \Pr \Omega = 1$.

It is written:


 * $X \sim \Poisson \lambda$

Also denoted as
Some sources denote this as:
 * $X \sim \map {\operatorname {Pois} } \lambda$

Some sources use $\mu$ instead of $\lambda$, but this can cause confusion with instances where $\mu$ is used for the expectation.

However, as the expectation of $\Poisson \lambda$ is also $\lambda$, this may not be as much of a confusion as all that.

Also see

 * Poisson Distribution Gives Rise to Probability Mass Function