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Raw Moment


A moment mu_n of a probability function P(x) taken about 0,

mu_n^'=<x^n>
(1)
=intx^nP(x)dx.
(2)

The raw moments mu_n^' (sometimes also called "crude moments") can be expressed as terms of the central moments mu_n (i.e., those taken about the mean mu) using the inverse binomial transform

 mu_n^'=sum_(k=0)^n(n; k)mu_kmu_1^('n-k),
(3)

with mu_0=1 and mu_1=0 (Papoulis 1984, p. 146). The first few values are therefore

mu_2^'=mu_2+mu_1^('2)
(4)
mu_3^'=mu_3+3mu_2mu_1^'+mu_1^('3)
(5)
mu_4^'=mu_4+4mu_3mu_1^'+6mu_2mu_1^('2)+mu_1^('4)
(6)
mu_5^'=mu_5+5mu_4mu_1^'+10mu_3mu_1^('2)+10mu_2mu_1^('3)+mu_1^('5).
(7)

The raw moments mu_n^' can also be expressed in terms of the cumulants kappa_n by exponentiating both sides of the series

 lnphi=ln(sum_(k=0)^infty((it)^k)/(k!)mu_k^')=sum_(n=0)^inftykappa_n((it)^n)/(n!),
(8)

where phi is the characteristic function, to obtain

 sum_(k=0)^infty((it)^k)/(k!)mu_k^'=exp(sum_(n=0)^inftykappa_n((it)^n)/(n!)).
(9)

The first few terms are then given by

mu_1^'=kappa_1
(10)
mu_2^'=kappa_1^2+kappa_2
(11)
mu_3^'=kappa_1^3+3kappa_1kappa_2+kappa_3
(12)
mu_4^'=kappa_1^4+6kappa_1^2kappa_2+3kappa_2^2+4kappa_1kappa_3+kappa_4
(13)
mu_5^'=kappa_1^5+10kappa_1^3kappa_2+15kappa_1kappa_2^2+10kappa_1^2kappa_3+10kappa_2kappa_3+5kappa_1kappa_4+kappa_5.
(14)

These transformations can be obtained using RawToCumulant[n] in the Mathematica application package mathStatica.

The raw moment of a multivariate probability function P(x_1,x_2,...) can be similarly defined as

 mu_(m,n,...)^'=<x_1^mx_2^n...>.
(15)

Therefore,

 mu_(n,0,...,0)^'=mu_n^'.
(16)

The multivariate raw moments can be expressed in terms of the multivariate cumulants. For example,

mu_(1,1)^'=kappa_(0,1)kappa_(1,0)+kappa_(1,1)
(17)
mu_(2,1)^'=kappa_(0,1)kappa_(1,0)^2+2kappa_(1,0)kappa_(1,1)+kappa_(0,1)kappa_(2,0)+kappa_(2,1).
(18)

These transformations can be obtained using RawToCumulant[{m, n, ...}] in the Mathematica application package mathStatica.


See also

Absolute Moment, Central Moment, Mean, Moment, Sample Raw Moment

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References

Kendall, M. G. "The Derivation of Multivariate Sampling Formulae from Univariate Formulae by Symbolic Operation." Ann. Eugenics 10, 392-402, 1940.Kenney, J. F. and Keeping, E. S. "Moments About the Origin." §7.2 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 91-92, 1962.Kratky, J.; Reinfelds, J.; Hutcheson, K.; and 47, L. R. "Tables of Crude Moments Expressed in Terms of Cumulants." Technical Report, University of Georgia, Athens, 1972.Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, 1984.Smith, P. J. "A Recursive Formulation of the Old Problem of Obtaining Moments from Cumulants and Vice Versa." Amer. Stat. 49, 217-218, 1995.

Referenced on Wolfram|Alpha

Raw Moment

Cite this as:

Weisstein, Eric W. "Raw Moment." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RawMoment.html

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