In the mathematical theory of
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
, the combinants ''c''
''n'' of a
random variable ''X'' are defined via the combinant-generating function ''G''(''t''), which is defined from the
moment generating function ''M''(''z'') as
:
which can be expressed directly in terms of a random variable ''X'' as
:
wherever this
expectation
Expectation or Expectations may refer to:
Science
* Expectation (epistemic)
* Expected value, in mathematical probability theory
* Expectation value (quantum mechanics)
* Expectation–maximization algorithm, in statistics
Music
* ''Expectation' ...
exists.
The ''n''th combinant can be obtained as the ''n''th derivatives of the logarithm of combinant generating function evaluated at –1 divided by ''n'' factorial:
:
Important features in common with the
cumulants are:
* the combinants share the additivity property of the cumulants;
* for
infinite divisibility (probability) distributions, both sets of moments are strictly positive.
References
*
Google Books
Theory of probability distributions
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