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In probability theory and statistics, a normal variance-mean mixture with mixing probability density g {\displaystyle g} is the continuous probability distribution of a random variable Y {\displaystyle Y} of the form
where α {\displaystyle \alpha } , β {\displaystyle \beta } and σ > 0 {\displaystyle \sigma >0} are real numbers, and random variables X {\displaystyle X} and V {\displaystyle V} are independent, X {\displaystyle X} is normally distributed with mean zero and variance one, and V {\displaystyle V} is continuously distributed on the positive half-axis with probability density function g {\displaystyle g}. The conditional distribution of Y {\displaystyle Y} given V {\displaystyle V} is thus a normal distribution with mean α + β V {\displaystyle \alpha +\beta V} and variance σ 2 V {\displaystyle \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 with drift β {\displaystyle \beta } and infinitesimal variance σ 2 {\displaystyle \sigma ^{2}} observed at a random time point independent of the Wiener process and with probability density function g {\displaystyle 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 {\displaystyle g} is
and its moment generating function is