Poisson distribution, a well-known discrete probability distribution, shows the likelihood of an event occurring within a specified time interval. These events are assumed to take place with a known average rate of success (given by the mean of the distribution) and independent of the last event.

If Random variable X, the number of events in a given interval, is distributed with mean (λ) and variance (σ² = λ), the probability of observing x events in n (many) trials when the probability of success λ /n is small, is obtained by:

P(X=x) = (e-λ * λx) / x!

, where x=0, 1, 2, 3, 4, … and P(X=x) is the Poisson probability and we say X follows a Poisson distribution with parameter λ.

The calculations provide either an exact probability or a cumulative probability, which refers to the probability that the Poisson random variable is greater or less than some specified limit. In contrast to a Binomial distribution that always has a finite upper bound, a Poisson random variable can take on any positive integer value. Poisson distribution has application in biological sciences, such as in predicting cell mutation within a large population.