Poisson distribution

The Poisson distribution is a probability distribution that models events that occur “completely at random” in time and space. We use these when we count the number of occurrences of an event (including a sequence of Bernoulli trials) in a certain time/space period.

The probability mass function for the Poisson RV is given by:

$$P(N=k)=\frac{{\alpha }^k}{k!}{e}^{-\alpha }$$

where $k\in {\mathbb{Z}}_{\ge 0}$, and $\alpha$ denotes the average number of event occurrences in a specified time interval or region in space.

Computations

The expected value of the Poisson distribution is:

$$E(N)=\alpha$$

And the variance is:

$${\sigma }_N^2=\alpha$$