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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“Quidquid luce fuit tenebris agit: but also the other way around. What we experience in dreams, so long as we experience it frequently, is in the end just as much a part of the total economy of our soul as anything we “really” experience: because of it we are richer or poorer, are sensitive to one need more or less, and are eventually guided a little by our dream-habits in broad daylight and even in the most cheerful moments occupying our waking spirit.”
—Friedrich Nietzsche (1844–1900)