Poisson refers to a term used in mathematics and statistics primarily to describe a specific type of probability distribution. The term "Poisson" can be attributed to a French mathematician, Simeon Denis Poisson, who developed the concept in the early 19th century.
In probability theory, the Poisson distribution is typically employed to characterize the number of events that occur within a fixed interval of time or space, given a constant average event rate and independence between events. It is commonly utilized to model rare and discrete events, such as the number of phone calls received per hour, the occurrences of accidents in a day, or the number of emails received during specific time intervals.
The Poisson distribution is characterized by a single parameter, often denoted as λ (lambda), which represents the average rate of event occurrence within the given time or space interval. This parameter determines both the mean and variance of the distribution, with the mean equal to λ and the variance also equal to λ. The shape of the distribution is determined by this average event rate, as higher values of λ result in a greater probability of observing higher values of events.
Mathematically, the probability mass function (PMF) of the Poisson distribution is defined by the formula P(x; λ) = (e^(-λ) * λ^x) / x!, where x represents the number of events, e is the base of the natural logarithm, and x! denotes the factorial of x.
Overall, the Poisson distribution serves as a useful tool in various fields, including mathematics, statistics, physics, biology, and economics, where the analysis of rare or discrete events is required.
