In probability theory and statistics, the generalized inverse Gaussian distribution (GIG) is a three-parameter family of continuous probability distributions with probability density function

where K_p is a modified Bessel function of the second kind, a > 0, b > 0 and p a real parameter. It is used extensively in geostatistics, statistical linguistics, finance, etc. This distribution was first proposed by Étienne Halphen. It was rediscovered and popularised by Ole Barndorff-Nielsen, who called it the generalized inverse Gaussian distribution. Its statistical properties are discussed in Bent Jørgensen's lecture notes.

By setting ${\displaystyle \theta ={\sqrt {ab}}}$ and ${\displaystyle \eta ={\sqrt {b/a}}}$, we can alternatively express the GIG distribution as

where ${\displaystyle \theta }$ is the concentration parameter while ${\displaystyle \eta }$ is the scaling parameter.

Barndorff-Nielsen and Halgreen proved that the GIG distribution is infinitely divisible.

The entropy of the generalized inverse Gaussian distribution is given as^{[citation needed]}

where ${\displaystyle \left[{\frac {d}{d\nu }}K_{\nu }\left({\sqrt {ab}}\right)\right]_{\nu =p}}$ is a derivative of the modified Bessel function of the second kind with respect to the order ${\displaystyle \nu }$ evaluated at ${\displaystyle \nu =p}$

The characteristic of a random variable ${\displaystyle X\sim GIG(p,a,b)}$ is given as (for a derivation of the characteristic function, see supplementary materials of )