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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 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.
The gamma distribution is a two-parameter exponential family with natural parameters α − 1 and −1/θ (equivalently, α − 1 and −λ), and natural statistics X and ln X. If the shape parameter α is held fixed, the resulting one-parameter family of distributions is a natural exponential family.
In probability and statistics, the class of exponential dispersion models (EDM), also called exponential dispersion family (EDF), is a set of probability distributions that represents a generalisation of the natural exponential family.
The Beta distribution on [0,1], a family of two-parameter distributions with one mode, of which the uniform distribution is a special case, and which is useful in estimating success probabilities. The four-parameter Beta distribution , a straight-forward generalization of the Beta distribution to arbitrary bounded intervals [ a , b ...
In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate; the distance parameter could be any meaningful mono-dimensional measure of the process, such as time ...
The Bernoulli distributions for form an exponential family. The maximum likelihood estimator of 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.
The inverse Gaussian distribution is a two-parameter exponential family with natural parameters −λ/(2μ 2) and −λ/2, and natural statistics X and 1/X.. For > fixed, it is also a single-parameter natural exponential family distribution [4] where the base distribution has density