In probability theory and statistics, a normal variance-mean mixture with mixing probability density g is the continuous probability distribution of a random variable Y of the form
Y = α + β V + σ V X , where α , β and σ > 0 are real numbers, and random variables X and V are independent, X is normally distributed with mean zero and variance one, and V is continuously distributed on the positive half-axis with probability density function g . The conditional distribution of Y given V is thus a normal distribution with mean α + β V and variance σ 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 β and infinitesimal variance σ 2 observed at a random time point independent of the Wiener process and with probability density function 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 g is
f ( x ) = ∫ 0 ∞ 1 2 π σ 2 v exp ( − ( x − α − β v ) 2 2 σ 2 v ) g ( v ) d v and its moment generating function is
M ( s ) = exp ( α s ) M g ( β s + 1 2 σ 2 s 2 ) , where M g is the moment generating function of the probability distribution with density function g , i.e.
M g ( s ) = E ( exp ( s V ) ) = ∫ 0 ∞ exp ( s v ) g ( v ) d v . 