Biomedical Engineering Reference
In-Depth Information
x
n
is defined as
μ
n
=
; where for Continuous Distributions (7.24):
f
(
x
)
=
f
(
x
)
P
(
x
)
d
x
(7.24)
and for Discrete Distributions (7.25)
f
(
x
)
P
(
x
)
f
(
x
)
=
(7.25)
μ
n
,
1
.
If the moment is taken about a point
a
rather the origin, the moment is termed
the second moment and is generated by (7.26).
the mean, is usually simply denoted as
μ
=
μ
(
x
=
(
x
a
)
n
=
a
)
n
P
(
n
)
μ
n
(
a
)
−
−
(7.26)
The moments are most commonly taken about the mean (a point
a
), which are also
referred to as “Central Moments” and are denoted as
μ
n
. The Central or second moment
about the origin is defined by (7.27), where
n
=
2.
μ
n
≡
(
x
)
n
=
)
n
P
(
x
)
dx
−
μ
(
x
−
μ
(7.27)
It should be noted that the third moment (
n
=
3) about the origin may also be called the
2
, which is equal to the variance or disbursement
of the distribution; and the square root of the variance is called the standard deviation.
second moment about the mean,
μ
=
σ
2
7.6 SUGGESTED READING
Papoulis, A.
Probability, Random Variables, and Stochastic Processes,
2nd ed. New York:
McGraw-Hill, pp. 145-149, 1984.
Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling, W. T. “Moments of a
Distribution: Mean, Variance, Skewness, and So Forth.”
14.1 in
Numerical Recipes
in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cam-
bridge University Press, pp. 604-609, 1992.
Kenney, J. F. and Keeping, E. S. “Moment-Generating and Characteristic Functions,”
“Some Examples of Moment-Generating Functions,” and “Uniqueness Theorem
§
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