Normal variance-mean mixture
where , and are real numbers, and random variables and are independent, is normally distributed with mean zero and variance one, and is continuously distributed on the positive half-axis with probability density function . The conditional distribution of given is thus a normal distribution with mean and variance . 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 and infinitesimal variance observed at a random time point independent of the Wiener process and with probability density function . 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 is
and its moment generating function is
where is the moment generating function of the probability distribution with density function , i.e.
O.E Barndorff-Nielsen, J. Kent and M. Sørensen (1982): "Normal variance-mean mixtures and z-distributions", International Statistical Review, 50, 145–159.