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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, including the enabling of the user to calculate expectations, covariances using differentiation based on some useful algebraic properties, as well as for generality, as exponential families are in a ...
The lognormal and Beta distribution are in the exponential family, but not the natural exponential family. The gamma distribution with two parameters is an exponential family but not a NEF and the chi-squared distribution is a special case of the gamma distribution with fixed scale parameter, and thus is also an exponential family but not a NEF ...
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The use of exponential generating functions (EGFs) to study the properties of Stirling numbers is a classical exercise in combinatorial mathematics and possibly the canonical example of how symbolic combinatorics is used. It also illustrates the parallels in the construction of these two types of numbers, lending support to the binomial-style ...
The Bernoulli distributions for form an exponential family. The maximum likelihood estimator of p {\displaystyle p} based on a random sample is the sample mean . The probability mass distribution function of a Bernoulli experiment along with its corresponding cumulative distribution function.
Exponential Random Graph Models (ERGMs) are a family of statistical models for analyzing data from social and other networks. [ 1 ] [ 2 ] Examples of networks examined using ERGM include knowledge networks, [ 3 ] organizational networks, [ 4 ] colleague networks, [ 5 ] social media networks, networks of scientific development, [ 6 ] and others.
In the univariate case, a real-valued random variable belongs to the additive exponential dispersion model with canonical parameter and index parameter , (,), if its probability density function can be written as
Parameter plane of the complex exponential family f(z)=exp(z)+c with 8 external ( parameter) rays. In the theory of dynamical systems, the exponential map can be used as the evolution function of the discrete nonlinear dynamical system. [1]