| has gloss | eng: In probability theory a normal variance-mean mixture with mixing probability density g is the continuous probability distribution of a random variable Y of the form :Y=\alpha + \beta V+\sigma \sqrtV}X, where \alpha and \beta are real numbers and \sigma > 0. The random variables X and V are independent, X is normal distributed with mean zero and variance one, and V is a continuous probability distribution on the positive half-axis with probability density function g. The conditional distribution of Y given V is thus a normal distribution with mean \alpha + \beta V and variance \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 \beta and infinitesimal variance \sigma^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. |