In probability and statistics, a natural exponential family (NEF) is a class of probability distributions that is a special case of an exponential family (EF). Definition Univariate case The natural exponential families (NEF) are a subset of the exponential families. A NEF is an exponential family in which the natural parameter η and the natural statistic T(x) are both the identity. A distribution in an exponential family with parameter θ can be written with probability density function (PDF) where and are known functions. A distribution in a natural exponential family with parameter θ can thus be written with PDF [Note that slightly different notation is used by the originator of the NEF, Carl Morris. Morris uses ω instead of η and ψ instead of A.] General multivariate case Suppose that , then a natural exponential family of order p has density or mass function of the form: where in this case the parameter Moment and cumulant generating functions A member of a natural exponential family has moment generating function (MGF) of the form The cumulant generating function is by definition the logarithm of the MGF, so it is Examples The five most important univariate cases are: normal distribution with known variance Poisson distribution gamma distribution with known shape parameter α (or k depending on notation set used) binomial distribution with known number of trials, n negative binomial distribution with known These five examples – Poisson, binomial, negative binomial, normal, and gamma – are a special subset of NEF, called NEF with quadratic variance function (NEF-QVF) because the variance can be written as a quadratic function of the mean. NEF-QVF are discussed below. Distributions such as the exponential, chi-squared, Bernoulli, and geometric distributions are special cases of the above five distributions. Many common distributions are either NEF or can be related to the NEF. For example: the chi-squared distribution is a special case of the gamma distribution. The Bernoulli distribution is a binomial distribution with n = 1 trial. The exponential distribution is a gamma distribution with shape parameter α = 1 (or k = 1 ). Some exponential family distributions are not NEF. The lognormal and Beta distribution are in the exponential family, but not the natural exponential family. The parameterization of most of the above distributions has been written differently from the parameterization commonly used in textbooks and the above linked pages. For example, the above parameterization differs from