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 if these events occur with a known constant mean rate and independently of the time since the last event. It is named after French mathematician Siméon Denis Poisson. The Poisson distribution is used to model situations where the number of occurrences of an event is rare, random, and independent of the occurrence of other events. Some key features of the Poisson distribution include:

• Probability mass function: The probability mass function of a Poisson distribution gives the probability of observing k events in a fixed interval of time or space, given a known constant mean rate of occurrence.

• Mean and variance: The mean and variance of a Poisson distribution are both equal to the parameter λ, which represents the mean rate of occurrence of events.

• Conditions for use: The Poisson distribution is used when the events occur independently of each other, the mean rate of occurrence is constant, and the probability of an event occurring in a given interval is proportional to the length of the interval.

The Poisson distribution is used in a variety of fields, including finance, economics, and physics, to model the occurrence of rare events. It is also used in Poisson regression and negative binomial regression to model count data.