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In probability and statistics, the generalized beta distribution [1] is a continuous probability distribution with four shape parameters, including more than thirty named distributions as limiting or special cases.
In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] or (0, 1) in terms of two positive parameters, denoted by alpha (α) and beta (β), that appear as exponents of the variable and its complement to 1, respectively, and control the shape of the distribution.
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 [,].
Note that if p = q = 1 then the generalized beta prime distribution reduces to the standard beta prime distribution. This generalization can be obtained via the following invertible transformation. If y ∼ β ′ ( α , β ) {\displaystyle y\sim \beta '(\alpha ,\beta )} and x = q y 1 / p {\displaystyle x=qy^{1/p}} for q , p > 0 {\displaystyle ...
Type IV subsumes the other types and is obtained when applying the logit transform to beta random variates. Following the same convention as for the log-normal distribution, type IV may be referred to as the logistic-beta distribution, with reference to the standard logistic function, which is the inverse of the logit transform.
The beta-binomial distribution is the binomial distribution in which the probability of success at each of n trials is not fixed but randomly drawn from a beta distribution. It is frequently used in Bayesian statistics, empirical Bayes methods and classical statistics to capture overdispersion in binomial type distributed data.
It is a multivariate generalization of the beta distribution, [1] hence its alternative name of multivariate beta distribution (MBD). [2] Dirichlet distributions are commonly used as prior distributions in Bayesian statistics , and in fact, the Dirichlet distribution is the conjugate prior of the categorical distribution and multinomial ...
G. Gamma distribution; Gamma/Gompertz distribution; Gaussian q-distribution; Generalised hyperbolic distribution; Generalized beta distribution; Generalized chi-squared distribution