The Gaussian probability density
function (PDF) (also referred to as the normal PDF)
for a scalar random variable xx is
defined as
∀x,x∈-∞∞:px=12πσ2ⅇ-12σ2x-μ2
x
x
p
x
1
2
σ
2
1
2
σ
2
x
μ
2
(1)
where
μμ is the mean and
σ2
σ
2
is the variance of
xx. It is denoted by
μσ2
μ
σ
2
and we say that
x∼μσ2
x
μ
σ
2
, where "∼" means "is distributed according
to."
The cumulative distribution function (CDF) for
μ=0
μ
0
and
σ2=1
σ
2
1
, for which the PDF is termed a standard
normal PDF, is defined as
Φx=∫-∞x12πⅇ-12t2dt
Φ
x
t
x
1
2
1
2
t
2
(2)
A more convenient description, which is termed the
right-tail probability and is the probability of
exceeding a given value, is defined as
Qx=1-Φx
Q
x
1
Φ
x
, where
Qx=∫x∞12πⅇ-12t2dt
Q
x
t
x
1
2
1
2
t
2
(3)
An approximation that is sometimes useful is
Qx≈12πxⅇ-12x2
Q
x
1
2
x
1
2
x
2
(4)
It is shown in Figure 1 along
with the exact value of
Qx
Q
x
. The approximation is quite accurate for
x>4
x
4
.
The multivariate Gaussian PDF of
an
n
×
1
n×1
random vector
x
x is defined as
px=12πn2detCⅇ-12x-μTC-1x-μ
p
x
1
2
n
2
C
1
2
x
μ
C
x
μ
(5)
where
μμ is the
mean vector and
CC
is the covariance matrix and is denoted by
μC
μ
C
. It is assumed that
CC is positive definite and hence
C-1
C
exists. The mean vector is defined as
∀i:i∈12…nμi=E
x
i
i
i
1
2
…
n
μ
i
x
i
(6)
and the covariance matrix as
∀ij,i∈12…n∧j∈12…n:Cij=E
x
i
-E
x
i
x
j
-E
x
j
i
j
i
1
2
…
n
j
1
2
…
n
C
i
j
x
i
x
i
x
j
x
j
(7)
or in more compact form
C=Ex-Exx-ExT
C
x
x
x
x
(8)
A chi-squared PDF with
νν degrees of freedom is defined
as
px=12ν2Γν2xν2-1ⅇ-12xifx>00ifx<0
p
x
1
2
ν
2
Γ
ν
2
x
ν
2
1
1
2
x
x
0
0
x
0
(9)
and is denoted by
χ
ν
2
χ
ν
2
. The degrees of freedom
νν is assumed to be an integer
with
ν≥1
ν
1
. The function
Γu
Γ
u
is the Gamma function which is defined as
Γu=∫0∞tu-1ⅇ-tdt
Γ
u
t
0
t
u
1
t
(10)
The relations
Γu=u-1Γu-1
Γ
u
u
1
Γ
u
1
for any
uu,
Γ12=π
Γ
1
2
, and
Γn=n-1!
Γ
n
n
1
for
nn an integer.
The chi-squared PDF arises as the PDF of
xx where
x=∑i=1ν
x
i
2
x
i
1
ν
x
i
2
if
x
i
∼01
x
i
0
1
and the
x
i
x
i
's are independent and identically distributed
(IID). By the latter we mean that each
x
i
x
i
is independent of the others and each
x
i
x
i
has the same PDF (identically distributed). The mean
and variance are
Ex=ν
x
ν
(11)
σx2=2ν
x
2
ν
(12)
A specific case of interest occurs when
ν=2
ν
2
so that
px=12ⅇ-12xifx>00ifx<0
p
x
1
2
1
2
x
x
0
0
x
0
(13)
and is referred to as an
exponential PDF (
Figure 2).
The F PDF arises as the ratio of two
independent
χ2
χ
2
random variables. Specifically, if
x=
x
1
ν
1
x
2
ν
2
x
x
1
ν
1
x
2
ν
2
(14)
where
x
1
∼
χ
ν
1
2
x
1
χ
ν
1
2
,
x
2
∼
χ
ν
2
2
x
2
χ
ν
2
2
, and
x
1
x
1
and
x 2
x 2
are independent, then
xx has the
F
PDF. It is given by
px=
ν
1
ν
2
ν
1
2B
ν
1
2
ν
2
2x
ν
1
2-11+
ν
1
ν
2
x
ν
1
+
ν
2
2ifx>00ifx<0
p
x
ν
1
ν
2
ν
1
2
B
ν
1
2
ν
2
2
x
ν
1
2
1
1
ν
1
ν
2
x
ν
1
ν
2
2
x
0
0
x
0
(15)
where
Buv
B
u
v
is the Beta function, which can be related to the
Gamma functions as
Buv=ΓuΓvΓu+v
B
u
v
Γ
u
Γ
v
Γ
u
v
(16)
The PDF is denoted by
F
ν
1
,
ν
2
F
ν
1
,
ν
2
as an
F PDF with
ν
1
ν
1
numerator degrees of freedom and
ν
2
ν
2
denominator degrees of freedom.
The mean and variance are
∀
ν
2
,
ν
2
>2:Ex=
ν
2
ν
2
-2
ν
2
ν
2
2
x
ν
2
ν
2
2
(17)
∀
ν
2
,
ν
2
>4:σx2=2
ν
2
2
ν
1
+
ν
2
-2
ν
1
ν
2
-22
ν
2
-4
ν
2
ν
2
4
x
2
ν
2
2
ν
1
ν
2
2
ν
1
ν
2
2
2
ν
2
4
(18)
