Examples of "cumulants"
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the moments of the distribution. The moments determine the cumulants in the sense that any two probability distributions whose moments are identical will have identical cumulants as well, and similarly the cumulants determine the moments.
The cumulants of the sum of the grouped variable and the uniform variable are the sums of the cumulants. As odd cumulants of a uniform distribution are zero; only even moments are affected.
If the moment-generating function does not exist, the cumulants can be defined in terms of the relationship between cumulants and moments discussed later.
These are "formal" cumulants and "formal" moments, as opposed to cumulants of a probability distribution and moments of a probability distribution.
The cumulants form a sequence of quantities such as mean, variance, skewness, kurtosis, and so on, that identify the distribution with increasing accuracy as more cumulants are used.
in free probability theory, the role of cumulants is occupied by "free cumulants", whose relation to ordinary cumulants is simply that the role of the set of all partitions of a finite set in the theory of ordinary cumulants is replaced by the set of all noncrossing partitions of a finite set. Just as the cumulants of degree more than 2 of a probability distribution are all zero if and only if the distribution is normal, so also, the "free" cumulants of degree more than 2 of a probability distribution are all zero if and only if the distribution is Wigner's semicircle distribution.
In a more advanced approach cross-cumulants are calculated by taking the information of several pixels into account. Cross-cumulants can be described as follows:
The combinatorial meaning of the expression of moments in terms of cumulants is easier to understand than that of cumulants in terms of moments:
Another important property of joint cumulants is multilinearity:
The lattice of noncrossing partitions plays the same role in defining free cumulants in free probability theory that is played by the lattice of "all" partitions in defining joint cumulants in classical probability theory. To be more precise, let formula_8 be a non-commutative probability space (See free probability for terminology.), formula_9 a non-commutative random variable with free cumulants formula_10. Then
The history of cumulants is discussed by Anders Hald.
It is most transparent when stated in its most general form, for "joint" cumulants, rather than for cumulants of a specified order for just one random variable. In general, we have
The law of total expectation and the law of total variance generalize naturally to conditional cumulants. The case "n" = 3, expressed in the language of (central) moments rather than that of cumulants, says
Important features in common with the cumulants are:
where the third cumulants are infinite, or as when
For higher cumulants, a simple and elegant generalization exists. See law of total cumulance.
For general 4th-order cumulants, the rule gives a sum of 15 terms, as follows:
The idea of cumulants was converted into quantum physics by Fritz Coester
The cumulants are obtained from a power series expansion of the cumulant generating function: