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:


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


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

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.