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Date: 16-2-2021
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The th raw moment (i.e., moment about zero) of a distribution is defined by
(1) |
where
(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
(4) |
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(5) |
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
(7) |
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(8) |
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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