Practical and incomplete statistics
- Definition 1: Mean
The mean function of a random process
X
t
X
t
is defined as the expected value of
X
t
X
t
for all
tt's.
μ
X
t
=E
X
t
={∫−∞∞xf
X
t
xd
x
if continuous∑
k
=−∞∞
x
k
p
X
t
x
k
if discrete
μ
X
t
X
t
x
x
f
X
t
x
continuous
k
x
k
p
X
t
x
k
discrete
(1)
- Definition 2: Autocorrelation
The autocorrelation function of the random process
X
t
X
t
is defined as
R
X
t
2
t
1
=E
X
t
2
X
t
1
¯={∫−∞∞∫−∞∞
x
2
x
1
¯f
X
t
2
X
t
1
x
2
x
1
d
x
1
d
x
2
if continuous∑
k
=−∞∞∑
l
=−∞∞
x
l
x
k
¯p
X
t
2
X
t
1
x
l
x
k
if discrete
R
X
t
2
t
1
X
t
2
X
t
1
x
2
x
1
x
2
x
1
f
X
t
2
X
t
1
x
2
x
1
continuous
k
l
x
l
x
k
p
X
t
2
X
t
1
x
l
x
k
discrete
(2)
If
X
t
X
t
is second-order stationary, then
R
X
t
2
t
1
R
X
t
2
t
1
only depends on
t
2
−
t
1
t
2
t
1
.
R
X
t
2
t
1
=E
X
t
2
X
t
1
¯=∫−∞∞∫−∞∞
x
2
x
1
¯f
X
t
2
X
t
1
x
2
x
1
d
x
2
d
x
1
R
X
t
2
t
1
X
t
2
X
t
1
x
1
x
2
x
2
x
1
f
X
t
2
X
t
1
x
2
x
1
(3)
R
X
t
2
t
1
=∫−∞∞∫−∞∞
x
2
x
1
¯f
X
t
2
-
t
1
X
0
x
2
x
1
d
x
2
d
x
1
=
R
X
t
2
−
t
1
0
R
X
t
2
t
1
x
1
x
2
x
2
x
1
f
X
t
2
-
t
1
X
0
x
2
x
1
R
X
t
2
t
1
0
(4)
If
R
X
t
2
t
1
R
X
t
2
t
1
depends on
t
2
−
t
1
t
2
t
1
only, then we will represent the autocorrelation with only one variable
τ=
t
2
−
t
1
τ
t
2
t
1
R
X
τ=
R
X
t
2
−
t
1
=
R
X
t
2
t
1
R
X
τ
R
X
t
2
t
1
R
X
t
2
t
1
(5)
-
R
X
0≥0
R
X
0
0
-
R
X
τ=
R
X
−τ¯
R
X
τ
R
X
τ
-
|
R
X
τ|≤
R
X
0
R
X
τ
R
X
0
X
t
=cos2π
f
0
t+Θω
X
t
2
f
0
t
Θ
ω
and
ΘΘ
is uniformly distributed between
00
and
2π
2
.
The mean function
μ
X
t=E
X
t
=Ecos2π
f
0
t+Θ=∫02πcos2π
f
0
t+θ12πd
θ
=0
μ
X
t
X
t
2
f
0
t
Θ
θ
0
2
2
f
0
t
θ
1
2
0
(6)
The autocorrelation function
R
X
t+τt=E
X
t
+
τ
X
t
¯=Ecos2π
f
0
(t+τ)+Θcos2π
f
0
t+Θ=1/2Ecos2π
f
0
τ+1/2Ecos2π
f
0
(2t+τ)+2Θ=1/2cos2π
f
0
τ+1/2∫02πcos2π
f
0
(2t+τ)+2θ12πd
θ
=1/2cos2π
f
0
τ
R
X
t
τ
t
X
t
+
τ
X
t
2
f
0
t
τ
Θ
2
f
0
t
Θ
12
2
f
0
τ
12
2
f
0
2
t
τ
2
Θ
12
2
f
0
τ
12
θ
0
2
2
f
0
2
t
τ
2
θ
1
2
12
2
f
0
τ
(7)
Not a function of
tt since the
second term in the right hand side of the equality in
Equation 7 is zero.
Toss a fair coin every TT
seconds. Since
X
t
X
t
is a discrete valued random process, the statistical characteristics
can be captured by the pmf and the mean function is written as
μ
X
t=E
X
t
=1/2×-1+1/2×1=0
μ
X
t
X
t
12
-1
12
1
0
(8)
R
X
t
2
t
1
=∑
k
k∑
l
l
x
k
x
l
p
X
t
2
X
t
1
x
k
x
l
=1×1×1/2−1×-1×1/2=1
R
X
t
2
t
1
k
k
l
l
x
k
x
l
p
X
t
2
X
t
1
x
k
x
l
1
1
12
-1
-1
12
1
(9)
when
nT≤
t
1
<(n+1)T
n
T
t
1
n
1
T
and
nT≤
t
2
<(n+1)T
n
T
t
2
n
1
T
R
X
t
2
t
1
=1×1×1/4−1×-1×1/4−1×1×1/4+1×-1×1/4=0
R
X
t
2
t
1
1
1
14
-1
-1
14
-1
1
14
1
-1
14
0
(10)
when
nT≤
t
1
<(n+1)T
n
T
t
1
n
1
T
and
mT≤
t
2
<(m+1)T
m
T
t
2
m
1
T
with
n≠m
n
m
R
X
t
2
t
1
={1 if (nT≤
t
1
<(n+1)T)∧(nT≤
t
2
<(n+1)T)0 otherwise
R
X
t
2
t
1
1
n
T
t
1
n
1
T
n
T
t
2
n
1
T
0
(11)
A function of
t
1
t
1
and
t
2
t
2
.
- Definition 3: Wide Sense Stationary
A process is said to be wide sense stationary if
μ
X
μ
X
is constant and
R
X
t
2
t
1
R
X
t
2
t
1
is only a function of
t
2
−
t
1
t
2
t
1
.
If
X
t
X
t
is strictly stationary, then it is wide sense stationary. The
converse is not necessarily true.
- Definition 4: Autocovariance
Autocovariance of a random process is defined as
C
X
t
2
t
1
=E(
X
t
2
−
μ
X
t
2
)
X
t
1
−
μ
X
t
1
¯=
R
X
t
2
t
1
−
μ
X
t
2
μ
X
t
1
¯
C
X
t
2
t
1
X
t
2
μ
X
t
2
X
t
1
μ
X
t
1
R
X
t
2
t
1
μ
X
t
2
μ
X
t
1
(12)
The variance of
X
t
X
t
is
Var
X
t
=
C
X
tt
Var
X
t
C
X
t
t
Two processes defined on one experiment (Figure 1).
- Definition 5: Crosscorrelation
The crosscorrelation function of a pair of random processes
is defined as
R
X
Y
t
2
t
1
=E
X
t
2
Y
t
1
¯=∫−∞∞∫−∞∞xyf
X
t
2
Y
t
1
xyd
x
d
y
R
X
Y
t
2
t
1
X
t
2
Y
t
1
y
x
x
y
f
X
t
2
Y
t
1
x
y
(13)
C
X
Y
t
2
t
1
=
R
X
Y
t
2
t
1
−
μ
X
t
2
μ
Y
t
1
¯
C
X
Y
t
2
t
1
R
X
Y
t
2
t
1
μ
X
t
2
μ
Y
t
1
(14)
- Definition 6: Jointly Wide Sense Stationary
The random processes
X
t
X
t
and
Y
t
Y
t
are said to be jointly wide sense stationary if
R
X
Y
t
2
t
1
R
X
Y
t
2
t
1
is a function of
t
2
−
t
1
t
2
t
1
only and
μ
X
t
μ
X
t
and
μ
Y
t
μ
Y
t
are constant.
