Let be the moment-generating function, then the cumulant generating function is given by

			(1)
			(2)

where , , ..., are the cumulants.

If

(3)

is a function of independent variables, then the cumulant-generating function for is given by

(4)

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.

CITE THIS AS:

Weisstein, Eric W. "Cumulant-Generating Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Cumulant-GeneratingFunction.html