Combinant
In the mathematical theory of probability, 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 :G_X(t)=M_X(\log(1+t)) which can be expressed directly in terms of a random variable ''X'' as : G_X(t) := E\left 1+t)^X\right \quad t \in \mathbb, 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: : c_n = \frac \frac \log(G (t)) \bigg, _ Important features in common with the cumulants are: * the combinants share the additivity property of the cumulan ...
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Probability Theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability ...
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