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Inverse gamma distribution is a special case of type 5 Pearson distribution; A multivariate generalization of the inverse-gamma distribution is the inverse-Wishart distribution. For the distribution of a sum of independent inverted Gamma variables see Witkovsky (2001)
The closely related inverse-gamma distribution is used as a conjugate prior for scale parameters, such as the variance of a normal distribution. If α is a positive integer, then the distribution represents an Erlang distribution; i.e., the sum of α independent exponentially distributed random variables, each of which has a mean of θ.
In probability theory and statistics, an inverse distribution is the distribution of the reciprocal of a random variable. Inverse distributions arise in particular in the Bayesian context of prior distributions and posterior distributions for scale parameters .
i.e., the inverse-gamma distribution, where () is the ordinary Gamma function. The Inverse Wishart distribution is a special case of the inverse matrix gamma distribution when the shape parameter = and the scale parameter =. Another generalization has been termed the generalized inverse Wishart distribution, .
In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy, this distribution, with frequency as the dependent variable, is known as a van der Waals profile. [note 1] It is a special case of the inverse-gamma distribution.
The quantile function can be found by noting that (;,,) = ((/)) where is the cumulative distribution function of the gamma distribution with parameters = / and =. The quantile function is then given by inverting F {\displaystyle F} using known relations about inverse of composite functions , yielding:
A Pearson density p is defined to be any valid solution to the differential equation (cf. Pearson 1895, p. 381) ′ () + + + + = ()with: =, = = +, =. According to Ord, [3] Pearson devised the underlying form of Equation (1) on the basis of, firstly, the formula for the derivative of the logarithm of the density function of the normal distribution (which gives a linear function) and, secondly ...
A generalization of this distribution which allows for a multivariate mean and a completely unknown positive-definite covariance matrix (whereas in the multivariate inverse-gamma distribution the covariance matrix is regarded as known up to the scale factor ) is the normal-inverse-Wishart distribution