In probability theory and statistics, a normal variance-mean mixture with mixing probability density
is the continuous probability distribution of a random variable
are real numbers, and random variables
is normally distributed with mean zero and variance one, and
is continuously distributed on the positive half-axis with probability density function
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
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
, 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.