Definition:Poisson Distribution

Definition

Let $X$ be a discrete random variable on a probability space $\left({\Omega, \Sigma, \Pr}\right)$.

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

• $\operatorname{Im} \left({X}\right) = \left\{{0, 1, 2, \ldots}\right\} = \N$

• $\displaystyle \Pr \left({X = k}\right) = \frac 1 {k!} \lambda^k e^{-\lambda}$

Note that Poisson Distribution Gives Rise to Probability Mass Function satisfying $\Pr \left({\Omega}\right) = 1$.

It is written:

$X \sim \operatorname{Pois} \left({\lambda}\right)$

or:

$X \sim \operatorname{Poisson} \left({\lambda}\right)$

Notational Differences

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 $\operatorname{Pois} \left({\lambda}\right)$ is also $\lambda$, this may not be as much of a confusion as all that.

Source of Name

This entry was named for Siméon-Denis Poisson.

Sources

• Geoffrey Grimmett: Probability: An Introduction (1986): $\S 2.2$