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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.
A gamma distribution with shape parameter α = 1 and rate parameter β is an exponential distribution with rate parameter β. A gamma distribution with shape parameter α = v/2 and rate parameter β = 1/2 is a chi-squared distribution with ν degrees of freedom.
The chi-squared distribution is a special case of the gamma distribution, in that (,) using the rate parameterization of the gamma distribution (or (,) using the scale parameterization of the gamma distribution) where k is an integer.
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]
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 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).