# factorial moment

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## factorial moment

[fak‚tȯr·ē·əl ′mō·mənt]
(statistics)
The n th factorial moment of a random variable X is the expected value of X (X-1) (X-2) ⋯ (X-n +1).
References in periodicals archive ?
Intermittency or factorial moment analysis applied to individual high multiplicity events may be used to identify events with densely populated narrow phase space regions.
Lianshou, "Single event 2-dimensional factorial moment analysis of [sup.197]Au-emulsion data from EMU01 experiment," Nuclear Physics B--Proceedings Supplements, vol.
It is not hard to see that, for r > 2, the r-th factorial moment of Xn is equal to the expected number of ordered r-tuples (G1,...
For every integer r > 2, the r-th factorial moment of [T.sub.k] is estimated as follows.
Also, the nth factorial moment of X with mean [lambda], denoted by [E.sub.[lambda]][[(X).sub.n]], is given by
The service times are independent and identically distributed with arbitrary distribution [{[s.sub.i]}.sup.[infinity].sub.i = 1] generating function S(x) = [[summation].sup.[infinity].sub.i = 1][s.sub.i][x.sup.i] and nth factorial moment [S.sub.n].
The vacation time is assumed to follow a general probability distribution [{[v.sub.i]}.sup.[infinity].sub.i = 1] with generating function V(x) = [[summation].sup.[infinity].sub.i = 1][v.sub.i][x.sup.i] and nth factorial moment [V.sub.n].
When the moments of the mixing variable exist, a necessary and sufficient condition that a distribution be a mixed Poisson is that its factorial moment (3) generating function be equal to the moment-generating function of the mixing distribution (Haight, 1967), namely
Theorem 3 The expectation and the second factorial moment of the number of superior elements that we reject when we consider the consecutive maxima instead of the true maxima, assuming random words of length n, are given by
The cancellations leave us with a second factorial moment of
X is said to be smaller than Y in factorial moments ordering, denoted by X [[less than or equal to].sub.fm] Y, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for all i [member of] {1,2...}.
This requires of course a very detailed description of the asymptotic behaviour of the ordinary or factorial moments of [A.sup.[H].sub.n] to take care of the huge cancellations when considering the centered random variable.
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