variables see Witkovsky (2001) Derivation from Gamma distribution Let , and recall that the pdf of the gamma distribution is , . Note that is the rate parameter from the perspective of the gamma distribution. Define the transformation . Then, the pdf of is Note that is the scale parameter from the perspective of the inverse gamma distribution. This can be straightforwardly demonstrated by seeing that satisfies the conditions for being a scale parameter. Occurrence Hitting time distribution of a Wiener process follows a Lévy distribution, which is a special case of the inverse-gamma distribution with . See also Gamma distribution Inverse-chi-squared distribution Normal distribution Pearson distribution References Hoff, P. (2009). "A first course in bayesian statistical methods". Springer. Continuous distributions Conjugate prior distributions Probability distributions with non-finite variance Exponential family distributions Gamma and related functions