Modeling With the Poisson Distribution

The Poisson distribution models the number of independent events occurring in a fixed interval (of time, length, area, etc.) when events happen at a constant average rate. If $X$ counts such events and the mean number of events in the interval is $\lambda,$ then

$$P(X = k) = \dfrac{e^{-\lambda}\, \lambda^k}{k!}, \qquad k = 0, 1, 2, \ldots$$

To model a situation, identify two things:

1. The count variable $X$ — what is being counted.
2. The rate parameter $\lambda$ — the expected number of events over the chosen interval.

For example, if a website averages $7$ visits per minute and $X$ is the number of visits in a particular minute, then $\lambda = 7$ and

$$P(X = 10) = \dfrac{e^{-7} \cdot 7^{10}}{10!} \approx 0.0710.$$