Define Poisson distribution.

The Poisson distribution is a probability distribution that models the number of rare events occurring in a fixed interval.

The Poisson distribution is named after French mathematician Siméon Denis Poisson, who first introduced it in 1837. It is used to model the number of rare events occurring in a fixed interval of time or space, such as the number of customers arriving at a store in an hour, the number of cars passing through a toll booth in a day, or the number of defects in a batch of products.

The Poisson distribution has a single parameter, λ (lambda), which represents the average number of events occurring in the interval. The probability of observing exactly k events in the interval is given by the Poisson probability mass function:

P(X=k) = (e^-λ * λ^k) / k!

where X is the random variable representing the number of events, e is the mathematical constant approximately equal to 2.71828, and k! is the factorial of k.

The mean and variance of the Poisson distribution are both equal to λ. This means that the distribution is symmetric and unimodal, with the peak at λ. As λ increases, the distribution becomes more spread out and skewed to the right.

The Poisson distribution is widely used in many fields, including physics, biology, finance, and engineering, to model rare events and estimate probabilities of their occurrence.