Poisson Distribution

Some experiments count how many times an event occurs over a fixed span of time, distance, area or volume — the number of car sales at a showroom in a day, or the number of customers arriving at a restaurant in an hour, for instance — rather than counting successes across a fixed number of trials. A discrete random variable of this kind follows a Poisson distribution when the events being counted satisfy a few conditions: they occur at a known, constant average rate; each occurrence is independent of how long it has been since the previous one; the rate of occurrence does not change with time; and the probability of an event happening is proportional to the length of the interval considered.

Let $X$ be the discrete random variable representing the number of occurrences of such an event over a fixed period. If $X$ follows a Poisson distribution, the probability of exactly $k$ occurrences is given by:

Here $e$ is Euler's number ($e \approx 2.71828$), $k$ is the number of occurrences, and $\lambda$ is a positive real number equal to both the mean and the variance of the distribution: $\lambda = E(X) = \mathrm{Var}(X)$. As required of any probability distribution, these probabilities sum to $1$ over all values of $k$. …