Poisson Distribution

Poisson distribution expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known constant mean rate and independently of the time since the last event. The notation is $\mathrm{Pois}(\lambda )$

Under a Poisson distribution with the expectation of $\lambda$ events in a given interval, the probability of $k$ events in the same interval (the PMF) is

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

where the positive real number $\lambda$ is equal to the expected value of $X$ and also to its variance:

$$\lambda =\mathrm{E}(X)=\mathrm{Var}(X)$$

Regenerative Property

Let $X_1\sim \mathrm{Pois}({\lambda }_1),X_2\sim \mathrm{Pois}({\lambda }_2)$ and $X_1,X_2$ are independent, then we will have

$$X_1+X_2=\mathrm{Pois}({\lambda }_1+{\lambda }_2)$$