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