The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, provided these events occur with a known constant mean rate and independently of the time since the last event. It is particularly useful for modeling count-based data, such as the number of customers arriving at a store in an hour or the number of service calls received in a day.

One of the key characteristics of the Poisson distribution is that it is defined by a single parameter, λ (lambda), which represents the average number of occurrences in the specified interval. The probability mass function of the Poisson distribution is given by the formula:

P(X = k) = (λ^k * e^(-λ)) / k!

where k is the number of occurrences, e is the base of the natural logarithm, and k! is the factorial of k.

The Poisson distribution is often compared to the binomial distribution, as both deal with counting occurrences. However, the Poisson distribution is more suitable for scenarios where the number of trials is large, and the probability of success is small ⁵.