Normal variance-mean mixture
In probability theory and statistics, a normal variance-mean mixture with mixing probability density formula_1 is the continuous probability distribution of a random variable formula_2 of the form where formula_4, formula_5 and formula_6 are real numbers, and random variables formula_7 and formula_8 are independent, formula_7 is normally distributed with mean zero and variance one, and formula_8 is continuously distributed on the positive half-axis with probability density function formula_1. The conditional distribution of formula_2 given formula_8 is thus a normal distribution with mean formula_14 and variance formula_15. 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 formula_5 and infinitesimal variance formula_17 observed at a random time point independent of the Wiener process and with probability density function formula_1. 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 formula_1 is and its moment generating function  is where formula_22 is the moment generating function of the probability distribution with density function formula_1, i.e. References. O.E Barndorff-Nielsen, J. Kent and M. Sørensen (1982): "Normal variance-mean mixtures and z-distributions", "International Statistical Review", 50, 145–159.