We get Moments of a pdf by setting
gX=Xn
g
X
X
n
in this previous equation,
EXn=∫xnfXxdx
X
n
x
x
n
fX
x
(1)
-
n=1
n
1
: 1st order moment,
Ex
x
= Mean value
-
n=2
n
2
: 2nd order moment,
Ex2
x
2
= Mean-squared value (Power or energy)
-
n>2
n
2
: Higher order moments,
Exn
x
n
, give more detail about
fXx
fX
x
.
Central moments are moments about the centre or
mean of a distribution,
EX−X¯n=∫x−X¯nfXxdx
X
X
n
x
x
X
n
fX
x
(2)
Some important parameters from central moments of a pdf are:
-
Variance,
n=2
n
2
:
σ2=EX−X¯2=∫x−X¯2fXxdx=∫x2fXxdx−2X¯∫xfXxdx+X¯2∫fXxdx=EX2−2X¯2+X¯2=EX2−X¯2
σ
2
X
X
2
x
x
X
2
fX
x
x
x
2
fX
x
2
X
x
x
fX
x
X
2
x
fX
x
X
2
2
X
2
X
2
X
2
X
2
(3)
-
Standard deviation,
σ=variance
σ
variance
.
-
Skewness,
n=3
n
3
:
γ=EX−X¯3σ3
γ
X
X
3
σ
3
(4)
γ=0
γ
0
if the pdf of XX is
symmetric about
X¯
X
, and becomes more positive if the tail of the
distribution is heavier when
X>X¯
X
X
.
-
Kurtosis (or excess),
n=4
n
4
:
κ=EX−X¯4σ4−3
κ
X
X
4
σ
4
3
(5)
κ=0
κ
0
for a Gaussian pdf and becomes more positive for
distributions with heavier tails.
Skewness and kurtosis are normalized by dividing the
central moments by appropriate powers of
σσ to make them
dimensionless. Kurtosis is usually offset by
-3-3 to make it zero for Gaussian
pdfs.
The normal (or Gaussian) pdf with zero mean is given by:
fXx=12πσ2ⅇ-x22σ2
fX
x
1
2
σ
2
x
2
2
σ
2
(6)
What is the
nnth order central
moment for the Gaussian?
Since the mean is zero, the nnth
order central moment is given by
EXn=∫xnfXxdx=12πσ2∫xnⅇ-x22σ2dx
X
n
x
x
n
fX
x
1
2
σ
2
x
x
n
x
2
2
σ
2
(7)
fXx
fX
x
is a function of
x2
x
2
and therefore is symmetric about zero. So all the
odd-order moments will integrate to zero (including the
lst-order moment, giving zero mean). The even-order moments
are then given by:
EXn=22πσ2∫0∞xnⅇ-x22σ2dx
X
n
2
2
σ
2
x
0
x
n
x
2
2
σ
2
(8)
where
nn is even. The integral is
calculated by substituting
u=x22σ2
u
x
2
2
σ
2
to give:
∫0∞xnⅇ-x22σ2dx=122σ2n+12∫0∞un−12ⅇ-udu=122σ2n+12Γn+12
x
0
x
n
x
2
2
σ
2
1
2
2
σ
2
n
1
2
u
0
u
n
1
2
u
1
2
2
σ
2
n
1
2
Γ
n
1
2
(9)
Here
Γz
Γ
z
is the Gamma function, which is defined as an
integral for all real
z>0
z
0
and is rather like the factorial function but
generalized to allow non-integer arguments. Values of the
Gamma function can be found in mathematical tables. It is
defined as follows:
Γz=∫0∞uz−1ⅇ-udu
Γ
z
u
0
u
z
1
u
(10)
and has the important (factorial-like) property that
∀z,z≠0:Γz+1=zΓz
z
z
0
Γ
z
1
z
Γ
z
(11)
∀z,z∈ℤ∧z>0:Γz+1=z!
z
z
z
0
Γ
z
1
z
(12)
The following results hold for the Gamma function (see below
for a way to evaluate
Γ12
Γ
1
2
etc.):
Γ12=π
Γ
1
2
(13)
Γ1=1
Γ
1
1
(14)
and hence
Γ32=π2
Γ
3
2
2
(15)
Γ2=1
Γ
2
1
(16)
Hence
EXn=0ifn=odd1π2σ2n2Γn+12ifn=even
X
n
0
n
odd
1
2
σ
2
n
2
Γ
n
1
2
n
even
(17)
-
Valid pdf,
n=0
n
0
:
EX0=1πΓ12=1
X
0
1
Γ
1
2
1
(18)
as required for a valid pdf.
The normalization factor
12πσ2
1
2
σ
2
in the expression for the pdf of a unit
variance Gaussian (e.g.
Equation 6) arises directly from the above result.
-
Mean,
n=1
n
1
:
EX=0
X
0
(19)
so the mean is zero.
-
Variance,
n=2
n
2
:
EX−X¯2=EX2=1π2σ2Γ32=1π2σ2π2=σ2
X
X
2
X
2
1
2
σ
2
Γ
3
2
1
2
σ
2
2
σ
2
(20)
Therefore standard deviation =
variance=σ
variance
σ
.
-
Skewness,
n=3
n
3
:
EX3=0
X
3
0
(21)
so the skewness is zero.
-
Kurtosis,
n=4
n
4
:
EX−X¯4=EX4=1π2σ22Γ52=1π2σ223π4=3σ4
X
X
4
X
4
1
2
σ
2
2
Γ
5
2
1
2
σ
2
2
3
4
3
σ
4
(22)
Hence
κ=EX−X¯4σ4−3=3−3=0
κ
X
X
4
σ
4
3
3
3
0
(23)
From the definition of ΓΓ
and substituting
u=x2
u
x
2
:
Γ12=∫0∞u-12ⅇ-udu=∫0∞x-1ⅇ-x22xdx=2∫0∞ⅇ-x2dx=∫-∞∞ⅇ-x2dx
Γ
1
2
u
0
u
1
2
u
x
0
x
-1
x
2
2
x
2
x
0
x
2
x
x
2
(24)
Using the following squaring trick to convert this to a 2-D
integral in polar coordinates:
Γ212=∫-∞∞ⅇ-x2dx∫-∞∞ⅇ-y2dy=∫-∞∞∫-∞∞ⅇ-x2+y2dxdy=∫-ππ∫0∞ⅇ-r2rdrdθ=2π-12ⅇ-r2|0∞=π
Γ
1
2
2
x
x
2
y
y
2
y
x
x
2
y
2
θ
r
0
r
2
r
2
0
1
2
r
2
(25)
and so (ignoring the negative square root):
Γ12=π≈1.7725
Γ
1
2
1.7725
(26)
Hence, using
Γz+1=zΓz
Γ
z
1
z
Γ
z
:
Γ32527292…=12π34π158π10516π…
Γ
3
2
5
2
7
2
9
2
…
1
2
3
4
15
8
105
16
…
(27)
The case for
z=1
z
1
is straightforward:
Γ1=∫0∞u0ⅇ-udu=-ⅇ-u|0∞=1
Γ
1
u
0
u
0
u
0
u
1
(28)
so
Γ2345…=12624…
Γ
2
3
4
5
…
1
2
6
24
…
(29)
