The Poisson distribution is a probability distribution that describes the probability of a given number of events occurring in a fixed interval of time or space, given that these events occur independently of the time since the last event. The Poisson distribution is often used in situations where the events occur at random, with a known average rate, and where the number of events that occur in any given interval of time or space is discrete (i.e., it can only take on integer values).
For example, the number of customers who arrive at a store in an hour, the number of cars that pass through a toll booth in a day, or the number of defects found in a batch of products could be modeled using a Poisson distribution.
The Poisson distribution is characterized by a single parameter, λ (lambda), which represents the average number of events that occur in the fixed interval. The probability of observing k events in the interval is given by the Poisson probability mass function:
P(k; λ) = (λ^k * e^(-λ)) / k!
where e is the mathematical constant approximately equal to 2.71828, and k! is the factorial of k.