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If the shape parameter of the gamma distribution is known, but the inverse-scale parameter is unknown, then a gamma distribution for the inverse scale forms a conjugate prior. The compound distribution, which results from integrating out the inverse scale, has a closed-form solution known as the compound gamma distribution. [22]
In practice the normal distribution is often parameterized in terms of the squared scale , which corresponds to the variance of the distribution. The gamma distribution is usually parameterized in terms of a scale parameter θ {\displaystyle \theta } or its inverse.
The formula in the definition of characteristic function allows us to compute φ when we know the distribution function F (or density f). If, on the other hand, we know the characteristic function φ and want to find the corresponding distribution function, then one of the following inversion theorems can be used. Theorem.
The generalized gamma distribution is a continuous probability distribution with two shape parameters (and a scale parameter).It is a generalization of the gamma distribution which has one shape parameter (and a scale parameter).
The reason for the usefulness of this characterization is that in Bayesian statistics the inverse gamma distribution is the conjugate prior distribution of the variance of a Gaussian distribution. As a result, the location-scale t distribution arises naturally in many Bayesian inference problems. [9]
The Rice distribution is a noncentral generalization of the Rayleigh distribution: () = (,). The Weibull distribution with the shape parameter k = 2 yields a Rayleigh distribution. Then the Rayleigh distribution parameter σ {\displaystyle \sigma } is related to the Weibull scale parameter according to λ = σ 2 . {\displaystyle \lambda =\sigma ...
Gaussian scale mixtures: [3] [4] Compounding a normal distribution with variance distributed according to an inverse gamma distribution (or equivalently, with precision distributed as a gamma distribution) yields a non-standardized Student's t-distribution. [5]
Also known as the (Moran-)Gamma Process, [1] the gamma process is a random process studied in mathematics, statistics, probability theory, and stochastics. The gamma process is a stochastic or random process consisting of independently distributed gamma distributions where N ( t ) {\displaystyle N(t)} represents the number of event occurrences ...