 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.