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Combinant

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Title: Combinant  
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Subject: Central moment, Quantile function, Cumulative distribution function, Moment (mathematics), Probability density function
Collection: Theory of Probability Distributions
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Combinant

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

G_X(t)=M_X(\log(1+t))

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

G_X(t) := E\left[(1+t)^X\right], \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:

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:

References

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