# Definition:Poisson Distribution

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## 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 and only if:

$\Img X = \set {0, 1, 2, \ldots} = \N$
$\map \Pr {X = k} = \dfrac 1 {k!} \lambda^k e^{-\lambda}$

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

• Results about the Poisson distribution can be found here.

## Source of Name

This entry was named for Siméon-Denis Poisson.

## Technical Note

The $\LaTeX$ code for $\Poisson {\lambda}$ is \Poisson {\lambda} .

When the argument is a single character, it is usual to omit the braces:

\Poisson \lambda