In probability theory, the inverse Gaussian distribution (also known as the Wald distribution) is a two-parameter family of continuous probability distributions with support on (0,∞).

Its probability density function is given by

for x > 0, where ${\displaystyle \mu >0}$ is the mean and ${\displaystyle \lambda >0}$ is the shape parameter.

The inverse Gaussian distribution has several properties analogous to a Gaussian distribution. The name can be misleading: it is an inverse only in that, while the Gaussian describes a Brownian motion's level at a fixed time, the inverse Gaussian describes the distribution of the time a Brownian motion with positive drift takes to reach a fixed positive level.

Its cumulant generating function (logarithm of the characteristic function)^{[contradictory]} is the inverse of the cumulant generating function of a Gaussian random variable.

To indicate that a random variable X is inverse Gaussian-distributed with mean μ and shape parameter λ we write ${\displaystyle X\sim \operatorname {IG} (\mu ,\lambda )\,\!}$.

The probability density function (pdf) of the inverse Gaussian distribution has a single parameter form given by

In this form, the mean and variance of the distribution are equal, ${\displaystyle \mathbb {E} [X]={\text{Var}}(X).}$