Varausjakauma

Varausjakauma, also known as 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 the French mathematician Siméon Denis Poisson.

The Poisson distribution is characterized by a single parameter, lambda (λ), which represents the average rate of occurrence of the event. The probability mass function of the Poisson distribution is given by:

P(X = k) = (e^(-λ) * λ^k) / k!

where X is the random variable representing the number of events, k is a non-negative integer, and e is the base of the natural logarithm.

The Poisson distribution has several key properties. It is a discrete distribution, meaning it can only take on non-negative integer values. It is also a unimodal distribution, meaning it has a single peak. The mean and variance of the Poisson distribution are both equal to λ.

The Poisson distribution is widely used in various fields, including statistics, engineering, and operations research. It is often used to model the number of events occurring in a fixed interval of time or space, such as the number of customers arriving at a store in an hour or the number of typos in a document. It is also used in quality control to model the number of defects in a product or the number of accidents in a workplace.