In probability theory and statistics, a normal variance-mean mixture with mixing probability density ${\displaystyle g}$ is the continuous probability distribution of a random variable ${\displaystyle Y}$ of the form

${\displaystyle Y=\alpha +\beta V+\sigma {\sqrt {V))X,}$

where ${\displaystyle \alpha }$, ${\displaystyle \beta }$ and ${\displaystyle \sigma >0}$ are real numbers, and random variables ${\displaystyle X}$ and ${\displaystyle V}$ are independent, ${\displaystyle X}$ is normally distributed with mean zero and variance one, and ${\displaystyle V}$ is continuously distributed on the positive half-axis with probability density function ${\displaystyle g}$. The conditional distribution of ${\displaystyle Y}$ given ${\displaystyle V}$ is thus a normal distribution with mean ${\displaystyle \alpha +\beta V}$ and variance ${\displaystyle \sigma ^{2}V}$. A normal variance-mean mixture can be thought of as the distribution of a certain quantity in an inhomogeneous population consisting of many different normal distributed subpopulations. It is the distribution of the position of a Wiener process (Brownian motion) with drift ${\displaystyle \beta }$ and infinitesimal variance ${\displaystyle \sigma ^{2))$ observed at a random time point independent of the Wiener process and with probability density function ${\displaystyle g}$. An important example of normal variance-mean mixtures is the generalised hyperbolic distribution in which the mixing distribution is the generalized inverse Gaussian distribution.

The probability density function of a normal variance-mean mixture with mixing probability density ${\displaystyle g}$ is

${\displaystyle f(x)=\int _{0}^{\infty }{\frac {1}{\sqrt {2\pi \sigma ^{2}v))}\exp \left({\frac {-(x-\alpha -\beta v)^{2)){2\sigma ^{2}v))\right)g(v)\,dv}$

and its moment generating function is

${\displaystyle M(s)=\exp(\alpha s)\,M_{g}\left(\beta s+{\frac {1}{2))\sigma ^{2}s^{2}\right),}$

where ${\displaystyle M_{g))$ is the moment generating function of the probability distribution with density function ${\displaystyle g}$, i.e.

${\displaystyle M_{g}(s)=E\left(\exp(sV)\right)=\int _{0}^{\infty }\exp(sv)g(v)\,dv.}$