The Poisson Distribution

The Poisson distribution models the number of independent events that occur in a fixed interval (of time, length, area, etc.) when those events happen at a known constant average rate.

If $X$ counts the events in one such interval and the average count is $\lambda > 0$, we write $X \sim \mathrm{Poisson}(\lambda)$. The probability mass function is

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

For example, if a help line receives calls at an average rate of $\lambda = 2$ per minute, the probability of receiving exactly $3$ calls in a given minute is

$$P(X = 3) = \dfrac{e^{-2} \cdot 2^3}{3!} = \dfrac{8\, e^{-2}}{6} = \dfrac{4\, e^{-2}}{3} \approx 0.180.$$

Notice that $k$ ranges over all non-negative integers, and the probabilities sum to $1$ since $\sum\limits_{k=0}^{\infty} \dfrac{\lambda^k}{k!} = e^{\lambda}$.