Let
be the moment-generating function,
then the cumulant generating function is given by
where ,
,
..., are the cumulants.
If
|
(3)
|
is a function of independent variables, then the cumulant-generating function
for
is given by
|
(4)
|
See also
Cumulant,
Moment-Generating
Function
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References
Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, p. 928, 1972.Kenney, J. F. and Keeping, E. S.
"Cumulants and the Cumulant-Generating Function" and "Additive Property
of Cumulants." §4.10-4.11 in Mathematics
of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 77-80,
1951.Referenced on Wolfram|Alpha
Cumulant-Generating Function
Cite this as:
Weisstein, Eric W. "Cumulant-Generating Function."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Cumulant-GeneratingFunction.html
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