Definition:Poisson Distribution

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 $$\Pr \left({\Omega}\right) = 1$$.