Definition 11/2

A discrete random variable taking on values 0,1,2,... with the probability mass function:

$$P(X=k)=\frac{\lambda^k e^{-\lambda}}{k!}$$

is called the Poisson distribution.

We can check it is a probability mass function because

$$\sum_{k=0}^{\infty} \frac{e^{-\lambda}\lambda^k}{k!}= e^{-\lambda}e^{\lambda}=1$$