Binomial type

:For other concepts using the name "binomial", see binomial (disambiguation).

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

Related Topics:
Cumulant

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: p_n'(0)=kappa_n= , the nth cumulant

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and

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: p_n(1)=mu_n'= , the nth moment.

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These are "formal" cumulants and "formal" moments, as opposed to cumulants of a probability distribution and moments of a probability distribution.

Related Topics:
Moments - Probability distribution

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Let

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:f(t)=sum_{n=1}^inftyrac{kappa_n}{n!}t^n

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be the (formal) cumulant-generating function. Then

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:f^{-1}(D) ,

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is the delta operator associated with the polynomial sequence, i.e., we have

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:f^{-1}(D)p_n(x)=np_{n-1}(x). ,

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