The random variable XX is said to
be a Gaussian random variable if its probability
density function has the form
pXx=12πσ2ⅇ-x-m22σ2
p
X
x
1
2
σ
2
x
m
2
2
σ
2
(1)
The mean of such a Gaussian random variable is
mm and its variance
σ2
σ
2
. As a shorthand notation, this information is denoted
by.
x∼mσ2
x
m
σ
2
. The characteristic function
Φ
X
·
Φ
X
·
of a Gaussian random variable is given by
Φ
X
ⅈu=ⅇⅈmuⅇ-σ2u22
Φ
X
u
m
u
σ
2
u
2
2
No closed form expression exists for the probability
distribution function of a Gaussian random variable. For a
zero-mean, unit-variance, Gaussian random variable
01
0
1
, the probability that it
exceeds the value
xx is denoted by
Qx
Q
x
.
PrX>x=1-PXx=12π∫x∞ⅇ-α22dα≡Qx
X
x
1
P
X
x
1
2
α
x
α
2
2
Q
x
A plot of
Q·
Q
·
is shown in
Figure 1. When the
Gaussian random variable has non-zero mean and/or non-unit
variance, the probability of it exceeding
xx can also be expressed in terms
of
Q·
Q
·
.
∀X,X∼mσ2:PrX>x=Qx-mσ
X
X
m
σ
2
X
x
Q
x
m
σ
(2)
Integrating by parts,
Q·
Q
·
is bounded (for
x>0
x
0
) by
12πx1+x2ⅇ-x22≤Qx≤12πxⅇ-x22
1
2
x
1
x
2
x
2
2
Q
x
1
2
x
x
2
2
(3)
As
xx becomes large, these bounds
approach each other and either can serve as an approximation to
Q·
Q
·
; the upper bound is usually chosen because of its
relative simplicity. The lower bound can be improved; noting
that the term
x1+x2
x
1
x
2
decreases for
x<1
x
1
and that
Qx
Q
x
increases as
xx
decreases, the term can be replaced by its value at
x=1
x
1
without affecting the sense of the bound for
x≤1
x
1
.
∀x,x≤1:122πⅇ-x22≤Qx
x
x
1
1
2
2
x
2
2
Q
x
(4)
We will have occasion to evaluate the expected value of
ⅇaX+bX2
a
X
b
X
2
where
X∼mσ2
X
m
σ
2
and
aa, bb
are constants. By definition,
EⅇaX+bX2=12πσ2∫-∞∞ⅇax+bx2-x-m22σ2dx
a
X
b
X
2
1
2
σ
2
x
a
x
b
x
2
x
m
2
2
σ
2
The argument of the exponential requires manipulation (i.e.,
completing the square) before the integral can be evaluated.
This expression can be written as
-12σ21-2bσ2x2-2m+aσ2x+m2
1
2
σ
2
1
2
b
σ
2
x
2
2
m
a
σ
2
x
m
2
Completing the square, this expression can be written
-1-2bσ22σ2x-m+aσ21-2bσ22+1-2bσ22σ2m+aσ21-2bσ22-m22σ2
1
2
b
σ
2
2
σ
2
x
m
a
σ
2
1
2
b
σ
2
2
1
2
b
σ
2
2
σ
2
m
a
σ
2
1
2
b
σ
2
2
m
2
2
σ
2
We are now ready to evaluate the integral. Using this expression,
EⅇaX+bX2=ⅇ1-2bσ22σ2m+aσ21-2bσ22-m22σ212πσ2∫-∞∞ⅇ-1-2bα22σ2x-m+aσ21-2bσ22dx
a
X
b
X
2
1
2
b
σ
2
2
σ
2
m
a
σ
2
1
2
b
σ
2
2
m
2
2
σ
2
1
2
σ
2
x
1
2
b
α
2
2
σ
2
x
m
a
σ
2
1
2
b
σ
2
2
Let
α=x-m+aσ21-2bσ2σ1-2bσ2
α
x
m
a
σ
2
1
2
b
σ
2
σ
1
2
b
σ
2
which implies that we must require that
1-2bσ2>0
1
2
b
σ
2
0
(or
b<12σ2
b
1
2
σ
2
). We then obtain
EⅇaX+bX2=ⅇ1-2bσ22σ2m+aσ21-2bσ22-m22σ211-2bσ212π∫-∞∞ⅇ-α22dα
a
X
b
X
2
1
2
b
σ
2
2
σ
2
m
a
σ
2
1
2
b
σ
2
2
m
2
2
σ
2
1
1
2
b
σ
2
1
2
α
α
2
2
The integral equals unity, leaving the result
∀b,b<12σ2:EⅇaX+bX2=ⅇ1-2bσ22σ2m+aσ21-2bσ22-m22σ21-2bσ2
b
b
1
2
σ
2
a
X
b
X
2
1
2
b
σ
2
2
σ
2
m
a
σ
2
1
2
b
σ
2
2
m
2
2
σ
2
1
2
b
σ
2
(5)
Important special cases are
-
a=0
a
0
,
X∼mσ2
X
m
σ
2
EⅇbX2=ⅇbm21-2bσ21-2bσ2
b
X
2
b
m
2
1
2
b
σ
2
1
2
b
σ
2
-
a=0
a
0
,
X∼0σ2
X
0
σ
2
EⅇbX2=11-2bσ2
b
X
2
1
1
2
b
σ
2
-
X∼0σ2
X
0
σ
2
EⅇaX+bX2=ⅇa2σ221-2bσ21-2bσ2
a
X
b
X
2
a
2
σ
2
2
1
2
b
σ
2
1
2
b
σ
2
The real-valued random vector X X is said to be a Gaussian
random vector if its joint distribution function has the
form
pXx=1det2πKⅇ-12x-mTK-1x-m
p
X
x
1
2
K
1
2
x
m
K
x
m
(6)
If complex-valued, the joint distribution of a circular Gaussian
random vector is given by
pXx=1detπKⅇ-x-mXT
K
X
-1x-mX
p
X
x
1
K
x
m
X
K
X
x
m
X
(7)
The vector
mX m
X denotes the
expected value of the Gaussian random vector and
KXKX
its covariance matrix.
mX=EX
m
X
X
K
X
=EXXT-mXmXT
K
X
X
X
m
X
m
X
As in the univariate case, the Gaussian distribution of a random
vector is denoted by
X∼mX
K
X
X
m
X
K
X
. After applying a linear transformation to Gaussian
random vector, such as
Y=AX
Y
A
X
, the result is also a Gaussian random vector (a random
variable if the matrix is a row vector):
Y∼AmXA
K
X
AT
Y
A
m
X
A
K
X
A
.
The characteristic function of a Gaussian random vector is given
by
Φ
X
ⅈu=ⅇ+ⅈuTmX-12uT
K
X
u
Φ
X
u
u
m
X
1
2
u
K
X
u
From this formula, the
N
th N
th -order moment formula
for jointly distributed Gaussian random variables is easily
derived.
E
X
1
…
X
N
=∑
𝒫
N
E
X
𝒫
N
(
1
)
X
𝒫
N
(
2
)
…E
X
𝒫
N
(
N
-
1
)
X
𝒫
N
(
N
)
ifN even∑
𝒫
N
E
X
𝒫
N
(
1
)
E
X
𝒫
N
(
2
)
X
𝒫
N
(
3
)
…E
X
𝒫
N
(
N
-
1
)
X
𝒫
N
(
N
)
ifN odd
X
1
…
X
N
𝒫
N
𝒫
N
X
𝒫
N
(
1
)
X
𝒫
N
(
2
)
…
X
𝒫
N
(
N
-
1
)
X
𝒫
N
(
N
)
N even
𝒫
N
𝒫
N
X
𝒫
N
(
1
)
X
𝒫
N
(
2
)
X
𝒫
N
(
3
)
…
X
𝒫
N
(
N
-
1
)
X
𝒫
N
(
N
)
N odd
where
𝒫 N
𝒫 N
denotes a permutation of the first
NN integers and
𝒫
N
i
𝒫
N
i
the
i
th
i
th
element of the permutation. For example,
E
X
1
X
2
X
3
X
4
=E
X
1
X
2
E
X
3
X
4
+E
X
1
X
3
E
X
2
X
4
+E
X
1
X
4
E
X
2
X
3
X
1
X
2
X
3
X
4
X
1
X
2
X
3
X
4
X
1
X
3
X
2
X
4
X
1
X
4
X
2
X
3
.
