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In the mathematical theory of probability, the combinants c_n of a random variable X are defined via the combinant-generating function G(t), which is defined from the moment generating function M(z) as

${\displaystyle G_{X}(t)=M_{X}(\log(1+t))}$

which can be expressed directly in terms of a random variable X as

${\displaystyle G_{X}(t):=E\left,\quad t\in \mathbb {R} ,}$

wherever this expectation exists.

The nth combinant can be obtained as the nth derivatives of the logarithm of combinant generating function evaluated at –1 divided by n factorial:

${\displaystyle c_{n}={\frac {1}{n!}}{\frac {\partial ^{n}}{\partial t^{n}}}\log(G(t)){\bigg |}_{t=-1}}$

Important features in common with the cumulants are:

• the combinants share the additivity property of the cumulants;
• for infinite divisibility (probability) distributions, both sets of moments are strictly positive.

References

• Kittel, W.; De Wolf, E. A. Soft Multihadron Dynamics. pp. 306 ff. ISBN 978-9812562951. Google Books

v t e Theory of probability distributions
probability mass function (pmf) probability density function (pdf) cumulative distribution function (cdf) quantile function
raw moment central moment mean variance standard deviation skewness kurtosis L-moment
moment-generating function (mgf) characteristic function probability-generating function (pgf) cumulant combinant

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