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 where \alpha, \beta and \sigma > 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 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. The probability density function of a normal variance-mean mixture with mixing probability density g is and its moment generating function is where M_g is the moment generating function of the probability distribution with density function g, i.e.