In probability theory and statistics, the cumulantsκn of a probability distribution are a set of quantities that provide an alternative to the moments of the distribution. Any two probability distributions whose moments are identical will have identical cumulants as well, and vice versa.
The first cumulant is the mean, the second cumulant is the variance, and the third cumulant is the same as the third central moment. But fourth and higher-order cumulants are not equal to central moments. In some cases theoretical treatments of problems in terms of cumulants are simpler than those using moments. In particular, when two or more random variables are statistically independent, the n-th-order cumulant of their sum is equal to the sum of their n-th-order cumulants. As well, the third and higher-order cumulants of a normal distribution are zero, and it is the only distribution with this property.
Just as for moments, where joint moments are used for collections of random variables, it is possible to define joint cumulants.
have identical cumulants as well, and vice versa. The first cumulant is the mean, the second cumulant is the variance, and the third cumulant is the same...
the third cumulant κ3 to the 1.5th power of the second cumulant κ2. This is analogous to the definition of kurtosis as the fourth cumulant normalized...
The Binder parameter or Binder cumulant in statistical physics, also known as the fourth-order cumulant U L = 1 − ⟨ s 4 ⟩ L 3 ⟨ s 2 ⟩ L 2 {\displaystyle...
from an autocorrelation function. One of the most common methods is the cumulant method, from which in addition to the sum of the exponentials above, more...
is κ, which is fine as long as it is clear that it does not refer to a cumulant. Other choices include γ2, to be similar to the notation for skewness,...
when n ≥ 4 is the nth cumulant κn(X). For n = 1, the nth cumulant is just the expected value; for n = either 2 or 3, the nth cumulant is just the nth central...
deviation skewness kurtosis L-moment moment-generating function (mgf) characteristic function probability-generating function (pgf) cumulant combinant...
positive level. Its cumulant generating function (logarithm of the characteristic function)[contradictory] is the inverse of the cumulant generating function...
mathematical statistics, the law of total cumulance is a generalization to cumulants of the law of total probability, the law of total expectation, and the...
natural log of the characteristic function, a new function is produced, the cumulant generating function, ln ( G ( k ) ) = ∑ m = 1 ∞ ( i k ) m m ! κ m , {\displaystyle...
obtain the cumulant function for different cases of the Tweedie models. A cumulant generating function (CGF) may then be obtained from the cumulant function...
bt}{bt}}.} For n ≥ 2 , {\displaystyle n\geq 2,} the n {\displaystyle n} -th cumulant of the continuous uniform distribution on the interval [ − 1 2 , 1 2 ]...
deviation skewness kurtosis L-moment moment-generating function (mgf) characteristic function probability-generating function (pgf) cumulant combinant...
before in Wigner's semi-circle law in the random matrix context. The free cumulant functional (introduced by Roland Speicher) plays a major role in the theory...
M_{X}(t)|_{t=\theta }} . It is easy to derive the cumulant-generation function of the proposal and therefore the proposal's cumulants. ψ θ ( η ) = log ( E θ exp (...
deviation skewness kurtosis L-moment moment-generating function (mgf) characteristic function probability-generating function (pgf) cumulant combinant...
might have suggested. Rather, their joint effect is to impart to Sufism a cumulant body of tradition, rather than individual and isolated experiences. In...
(X)=\operatorname {Cov} (X,X).} The variance is also equivalent to the second cumulant of a probability distribution that generates X {\displaystyle X} . The...
deviation skewness kurtosis L-moment moment-generating function (mgf) characteristic function probability-generating function (pgf) cumulant combinant...
[citation needed] The normal distribution is the only distribution whose cumulants beyond the first two (i.e., other than the mean and variance) are zero...
When λ is a positive integer, the modes are λ and λ − 1. All of the cumulants of the Poisson distribution are equal to the expected value λ. The n th...
deviation skewness kurtosis L-moment moment-generating function (mgf) characteristic function probability-generating function (pgf) cumulant combinant...
An inequality on location and scale parameters Coefficient of variation Cumulant Deviation (statistics) Distance correlation Distance standard deviation...
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