Biomedical Engineering Reference
In-Depth Information
the probability of the occurrence of the given realization
p
(
x
k
,
t
1
)
:
N
k
=
1
x
k
(
t
1
)
p
(
x
k
,
t
1
)
μ
x
(
t
1
)=
E
[
ξ
(
t
1
)] =
lim
N
(1.2)
→
∞
E
[
.
]
denotes expected value. In general the expected value of the given function
f
(
ξ
)
may be expressed by:
N
k
=
1
f
(
x
k
(
t
1
))
p
(
x
k
,
t
1
)
E
[
f
(
ξ
(
t
1
))] =
lim
N
(1.3)
→
∞
If the probability of occurrence of each realization is the same, which frequently
is the case, the equation (1.3) is simplified:
N
k
=
1
f
(
x
k
(
t
1
))
1
N
[
(
(
))] =
E
f
ξ
t
1
lim
N
(1.4)
→
∞
In particular, function
f
(
ξ
)
can represent moments or joint moments of the pro-
ξ
n
. In these terms mean value (1.2) is
a first order moment and mean square value ψ
2
cesses ξ
(
t
)
. Moment of order
n
is :
f
(
ξ
)=
is the second order moment of the
process:
E
ξ
2
)
=
N
k
=
1
x
k
(
t
1
)
p
(
x
k
,
t
1
)
ψ
2
(
t
1
)=
(
t
1
lim
N
(1.5)
→
∞
Central moments
m
n
about the mean are calculated in respect to the mean value
μ
x
.Thefirst central moment is zero. The second order central moment is variance:
E
2
σ
x
=
=
(
−
)
m
2
ξ
μ
x
(1.6)
where σ is the standard deviation. The third order central moment in an analogous
way is defined as:
E
3
=
(
−
)
m
3
ξ
μ
x
(1.7)
Parameter β
1
related to
m
3
:
m
3
m
3
/
2
=
m
3
σ
3
β
1
=
(1.8)
is called skewness, since it is equal to 0 for symmetric probability distributions of
p
(
x
k
,
.
Kurtosis:
t
1
)
m
4
m
2
β
2
=
(1.9)
isameasureofflatness of the distribution. For normal distribution kurtosis is equal to
3. A high kurtosis distribution has a sharper peak and longer, fatter tails, in contrast to
a low kurtosis distribution which has a more rounded peak and shorter thinner tails.
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