Moment
المؤلف:
Papoulis, A.
المصدر:
Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill
الجزء والصفحة:
...
22-2-2021
1994
Moment
The 
th raw moment 
(i.e., moment about zero) of a distribution 
is defined by
 |
(1)
|
where
![<f(x)>=<span style=]() {sumf(x)P(x) discrete distribution; intf(x)P(x)dx continuous distribution. " src="https://mathworld.wolfram.com/images/equations/Moment/NumberedEquation2.gif" style="height:68px; width:306px" /> |
(2)
|
, the mean, is usually simply denoted 
. If the moment is instead taken about a point 
,
 |
(3)
|
A statistical distribution is not uniquely specified by its moments, although it is by its characteristic function.
The moments are most commonly taken about the mean. These so-called central moments are denoted 
and are defined by
with 
. The second moment about the mean is equal to the variance
 |
(6)
|
where 
is called the standard deviation.
The related characteristic function is defined by
The moments may be simply computed using the moment-generating function,
 |
(9)
|
REFERENCES:
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: Cambridge University Press, pp. 604-609, 1992.
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