Cumulants and Moments
The sequence κn of coefficients of the first-degree terms in a polynomial sequence of binomial type may be termed the cumulants of the polynomial sequence. It can be shown that the whole polynomial sequence of binomial type is determined by its cumulants, in a way discussed in the article titled cumulant. Thus
- the nth cumulant
and
- the nth moment.
These are "formal" cumulants and "formal" moments, as opposed to cumulants of a probability distribution and moments of a probability distribution.
Let
be the (formal) cumulant-generating function. Then
is the delta operator associated with the polynomial sequence, i.e., we have
Read more about this topic: Binomial Type
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