Normal variance-mean mixture
In probability theory and statistics, a normal variance-mean mixture with mixing probability density is the continuous probability distribution of a random variable of the form
where, and are real numbers, and random variables and are independent, is normally distributed with mean zero and variance one, and is continuously distributed on the positive half-axis with probability density function. The conditional distribution of given is thus a normal distribution with mean and variance. 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 and infinitesimal variance 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 is the moment generating function of the probability distribution with density function, i.e.
