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
=
α
+
β
V
+
σ
V
X
,
where
α
,
β
and
σ
>
0
are real numbers, 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
α
+
β
V
and variance
σ
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
β
and infinitesimal variance
σ
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
)
=
∫
0
∞
1
2
π
σ
2
v
exp
(
−
(
x
−
α
−
β
v
)
2
2
σ
2
v
)
g
(
v
)
d
v
and its moment generating function is
M
(
s
)
=
exp
(
α
s
)
M
g
(
β
s
+
1
2
σ
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
(
s
V
)
)
=
∫
0
∞
exp
(
s
v
)
g
(
v
)
d
v
.
