Komar mass

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

$Y=\alpha +\beta V+\sigma \sqrt{V}X,$

where $\alpha$ and $\beta$ are real numbers and $\sigma >0$ 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 $\alpha +\beta V$ 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

$f(x)=\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{1}{\sqrt{2\pi {\sigma }^{2}v}}\exp \left(\frac{-(x-\alpha -\beta v{)}^2}{2{\sigma }^{2}v}\right)g(v)dv$

and its moment generating function is

$M(s)=\exp (\alpha s)M_g(\beta s+\frac{1}{2}{\sigma }^2{s}^2),$

where $M_g$ is the moment generating function of the probability distribution with density function $g$, i.e.

$M_g(s)=E(\exp (sV))=\underset{0}{\overset{\mathrm{\infty }}{\int }}\exp (sv)g(v)dv.$

See also

• Normal-inverse Gaussian distribution

References

O.E Barndorff-Nielsen, J. Kent and M. Sørensen (1982): "Normal variance-mean mixtures and z-distributions", International Statistical Review, 50, 145–159.