Poissoneloszlást

Poissoneloszlást, also known as 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 was developed by French mathematician Siméon Denis Poisson in 1837.

The Poisson distribution is often used to model the number of times an event occurs within a fixed interval of time or space. For example, it can be used to model the number of customers entering a store in an hour, the number of emails received by a server in a day, or the number of defects in a manufactured product.

The probability mass function of a Poisson distribution is given by:

P(X = k) = (λ^k * e^(-λ)) / k!

where k is the number of events, λ is the average rate of occurrence, e is the base of the natural logarithm, and k! is the factorial of k.

The Poisson distribution has one parameter, λ, which is the average rate of occurrence. The mean and variance of a Poisson distribution are both equal to λ.

The Poisson distribution is a special case of the binomial distribution, where the number of trials is infinite and the probability of success is very small. It is also related to the exponential distribution, which models the time between events in a Poisson process.

The Poisson distribution is widely used in various fields such as physics, engineering, biology, and finance. It is a fundamental concept in queueing theory, reliability theory, and risk assessment.