# Exponential family

In probability and statistics, an **exponential family** is a parametric set of probability distributions of a certain form, specified below. This special form is chosen for mathematical convenience, based on some useful algebraic properties, as well as for generality, as exponential families are in a sense very natural sets of distributions to consider. The term **exponential class** is sometimes used in place of "exponential family",[1] or the older term **Koopman–Darmois family**. The terms "distribution" and "family" are often used loosely: properly, *an* exponential family is a *set* of distributions, where the specific distribution varies with the parameter;[lower-alpha 1] however, a parametric *family* of distributions is often referred to as "*a* distribution" (like "the normal distribution", meaning "the family of normal distributions"), and the set of all exponential families is sometimes loosely referred to as "the" exponential family. They are distinct because they possess a variety of desirable properties, most importantly the existence of a sufficient statistic.

The concept of exponential families is credited to[2] E. J. G. Pitman,[3] G. Darmois,[4] and B. O. Koopman[5] in 1935–1936. Exponential families of distributions provides a general framework for selecting a possible alternative parameterisation of a parametric family of distributions, in terms of **natural parameters**, and for defining useful sample statistics, called the **natural sufficient statistics** of the family.