Combinant

Mathematical theory

This article relies largely or entirely on a single source. Relevant discussion may be found on the talk page. Please help improve this article by introducing citations to additional sources.
Find sources: "Combinant" – news · newspapers · books · scholar · JSTOR (March 2024)

|

In the mathematical theory of probability, the combinants cn 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

G X ( t ) = M X ( log ⁡ ( 1 + t ) ) {\displaystyle 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 [ ( 1 + t ) X ] , t ∈ R , {\displaystyle 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 = 1 n ! ∂ n ∂ t n log ⁡ ( G ( t ) ) | t = − 1 {\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

[edit]

* 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

This probability-related article is a stub. You can help Wikipedia by expanding it.

|

* v
* t
* e

*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template