Normal variance-mean mixture
Encyclopedia
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 
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
(Brownian motion) 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.
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
and statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, a normal variance-mean mixture with mixing probability density
is the continuous probability distribution of a random variable of the formwhere
and are real numbers and 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 functionProbability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...
. The conditional distributionConditional distribution
Given two jointly distributed random variables X and Y, the conditional probability distribution of Y given X is the probability distribution of Y when X is known to be a particular value...
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 processWiener process
In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown...
(Brownian motion) 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
Mixture density
In probability and statistics, a mixture distribution is the probability distribution of a random variable whose values can be interpreted as being derived in a simple way from an underlying set of other random variables. In particular, the final outcome value is selected at random from among the...
isand its moment generating function is
where
is the moment generating function of the probability distribution with density function , i.e.