The normal-inverse Gaussian distribution (NIG, also known as the normal-Wald distribution) is a continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the inverse Gaussian distribution. The NIG distribution was noted by Blaesild in 1977 as a subclass of the generalised hyperbolic distribution discovered by Ole Barndorff-Nielsen. In the next year Barndorff-Nielsen published the NIG in another paper. It was introduced in the mathematical finance literature in 1997.

The parameters of the normal-inverse Gaussian distribution are often used to construct a heaviness and skewness plot called the NIG-triangle.

The fact that there is a simple expression for the moment generating function implies that simple expressions for all moments are available.

This class is closed under affine transformations, since it is a particular case of the Generalized hyperbolic distribution, which has the same property. If

then

This class is infinitely divisible, since it is a particular case of the Generalized hyperbolic distribution, which has the same property.

The class of normal-inverse Gaussian distributions is closed under convolution in the following sense: if ${\displaystyle X_{1}}$ and ${\displaystyle X_{2}}$ are independent random variables that are NIG-distributed with the same values of the parameters ${\displaystyle \alpha }$ and ${\displaystyle \beta }$, but possibly different values of the location and scale parameters, ${\displaystyle \mu _{1}}$, ${\displaystyle \delta _{1}}$ and ${\displaystyle \mu _{2},}$ ${\displaystyle \delta _{2}}$, respectively, then ${\displaystyle X_{1}+X_{2}}$ is NIG-distributed with parameters ${\displaystyle \alpha ,}$ ${\displaystyle \beta ,}$${\displaystyle \mu _{1}+\mu _{2}}$ and ${\displaystyle \delta _{1}+\delta _{2}.}$

The class of NIG distributions is a flexible system of distributions that includes fat-tailed and skewed distributions, and the normal distribution, ${\displaystyle N(\mu ,\sigma ^{2}),}$ arises as a special case by setting ${\displaystyle \beta =0,\delta =\sigma ^{2}\alpha ,}$ and letting ${\displaystyle \alpha \rightarrow \infty }$.