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In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to the gamma distribution.
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.
Inverse distributions arise in particular in the Bayesian context of prior distributions and posterior distributions for scale parameters. In the algebra of random variables, inverse distributions are special cases of the class of ratio distributions, in which the numerator random variable has a degenerate distribution.
The scaled-inverse-chi-squared distribution is exactly the same distribution as the inverse gamma distribution, but with a different parameterization, i.e. = , = . 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 ...
Ordinary least squares regression of Okun's law.Since the regression line does not miss any of the points by very much, the R 2 of the regression is relatively high.. In statistics, the coefficient of determination, denoted R 2 or r 2 and pronounced "R squared", is the proportion of the variation in the dependent variable that is predictable from the independent variable(s).
In probability and statistics, the inverse-chi-squared distribution (or inverted-chi-square distribution [1]) is a continuous probability distribution of a positive-valued random variable. It is closely related to the chi-squared distribution. It is used in Bayesian inference as conjugate prior for the variance of the normal distribution. [2]
Usually, the inverse gamma function refers to the principal branch with domain on the real interval [, +) and image on the real interval [, +), where = … [2] is the minimum value of the gamma function on the positive real axis and = = … [3] is the location of that minimum.
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, .